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Theorem txkgen 23643
Description: The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on 𝑆 can also be replaced with either "compactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
txkgen ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ ran 𝑘Gen)

Proof of Theorem txkgen
Dummy variables 𝑎 𝑏 𝑘 𝑠 𝑡 𝑢 𝑥 𝑦 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 23464 . . 3 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
2 elinel1 4195 . . . 4 (𝑆 ∈ (ran 𝑘Gen ∩ Haus) → 𝑆 ∈ ran 𝑘Gen)
3 kgentop 23533 . . . 4 (𝑆 ∈ ran 𝑘Gen → 𝑆 ∈ Top)
42, 3syl 17 . . 3 (𝑆 ∈ (ran 𝑘Gen ∩ Haus) → 𝑆 ∈ Top)
5 txtop 23560 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
61, 4, 5syl2an 594 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ Top)
7 simplll 773 . . . . . . . 8 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑅 ∈ 𝑛-Locally Comp)
8 eqid 2726 . . . . . . . . . 10 (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) = (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩)
98mptpreima 6241 . . . . . . . . 9 ((𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) “ 𝑥) = {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥}
101ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑅 ∈ Top)
11 toptopon2 22907 . . . . . . . . . . . . . 14 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
1210, 11sylib 217 . . . . . . . . . . . . 13 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑅 ∈ (TopOn‘ 𝑅))
13 idcn 23248 . . . . . . . . . . . . 13 (𝑅 ∈ (TopOn‘ 𝑅) → ( I ↾ 𝑅) ∈ (𝑅 Cn 𝑅))
1412, 13syl 17 . . . . . . . . . . . 12 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ( I ↾ 𝑅) ∈ (𝑅 Cn 𝑅))
15 simpllr 774 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑆 ∈ (ran 𝑘Gen ∩ Haus))
1615, 4syl 17 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑆 ∈ Top)
17 toptopon2 22907 . . . . . . . . . . . . . 14 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
1816, 17sylib 217 . . . . . . . . . . . . 13 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑆 ∈ (TopOn‘ 𝑆))
19 simpr 483 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑦𝑥)
20 simplr 767 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆)))
21 elunii 4912 . . . . . . . . . . . . . . . 16 ((𝑦𝑥𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → 𝑦 (𝑘Gen‘(𝑅 ×t 𝑆)))
2219, 20, 21syl2anc 582 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑦 (𝑘Gen‘(𝑅 ×t 𝑆)))
23 eqid 2726 . . . . . . . . . . . . . . . . . 18 𝑅 = 𝑅
24 eqid 2726 . . . . . . . . . . . . . . . . . 18 𝑆 = 𝑆
2523, 24txuni 23583 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
2610, 16, 25syl2anc 582 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
2710, 16, 5syl2anc 582 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑅 ×t 𝑆) ∈ Top)
28 eqid 2726 . . . . . . . . . . . . . . . . . 18 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2928kgenuni 23530 . . . . . . . . . . . . . . . . 17 ((𝑅 ×t 𝑆) ∈ Top → (𝑅 ×t 𝑆) = (𝑘Gen‘(𝑅 ×t 𝑆)))
3027, 29syl 17 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑅 ×t 𝑆) = (𝑘Gen‘(𝑅 ×t 𝑆)))
3126, 30eqtrd 2766 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ( 𝑅 × 𝑆) = (𝑘Gen‘(𝑅 ×t 𝑆)))
3222, 31eleqtrrd 2829 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑦 ∈ ( 𝑅 × 𝑆))
33 xp2nd 8027 . . . . . . . . . . . . . 14 (𝑦 ∈ ( 𝑅 × 𝑆) → (2nd𝑦) ∈ 𝑆)
3432, 33syl 17 . . . . . . . . . . . . 13 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (2nd𝑦) ∈ 𝑆)
35 cnconst2 23274 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆) ∧ (2nd𝑦) ∈ 𝑆) → ( 𝑅 × {(2nd𝑦)}) ∈ (𝑅 Cn 𝑆))
3612, 18, 34, 35syl3anc 1368 . . . . . . . . . . . 12 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ( 𝑅 × {(2nd𝑦)}) ∈ (𝑅 Cn 𝑆))
37 fvresi 7178 . . . . . . . . . . . . . . . 16 (𝑡 𝑅 → (( I ↾ 𝑅)‘𝑡) = 𝑡)
38 fvex 6905 . . . . . . . . . . . . . . . . 17 (2nd𝑦) ∈ V
3938fvconst2 7212 . . . . . . . . . . . . . . . 16 (𝑡 𝑅 → (( 𝑅 × {(2nd𝑦)})‘𝑡) = (2nd𝑦))
4037, 39opeq12d 4881 . . . . . . . . . . . . . . 15 (𝑡 𝑅 → ⟨(( I ↾ 𝑅)‘𝑡), (( 𝑅 × {(2nd𝑦)})‘𝑡)⟩ = ⟨𝑡, (2nd𝑦)⟩)
4140mpteq2ia 5248 . . . . . . . . . . . . . 14 (𝑡 𝑅 ↦ ⟨(( I ↾ 𝑅)‘𝑡), (( 𝑅 × {(2nd𝑦)})‘𝑡)⟩) = (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩)
4241eqcomi 2735 . . . . . . . . . . . . 13 (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) = (𝑡 𝑅 ↦ ⟨(( I ↾ 𝑅)‘𝑡), (( 𝑅 × {(2nd𝑦)})‘𝑡)⟩)
4323, 42txcnmpt 23615 . . . . . . . . . . . 12 ((( I ↾ 𝑅) ∈ (𝑅 Cn 𝑅) ∧ ( 𝑅 × {(2nd𝑦)}) ∈ (𝑅 Cn 𝑆)) → (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)))
4414, 36, 43syl2anc 582 . . . . . . . . . . 11 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) ∈ (𝑅 Cn (𝑅 ×t 𝑆)))
45 llycmpkgen 23543 . . . . . . . . . . . . 13 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ ran 𝑘Gen)
4645ad3antrrr 728 . . . . . . . . . . . 12 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑅 ∈ ran 𝑘Gen)
476ad2antrr 724 . . . . . . . . . . . 12 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑅 ×t 𝑆) ∈ Top)
48 kgencn3 23549 . . . . . . . . . . . 12 ((𝑅 ∈ ran 𝑘Gen ∧ (𝑅 ×t 𝑆) ∈ Top) → (𝑅 Cn (𝑅 ×t 𝑆)) = (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆))))
4946, 47, 48syl2anc 582 . . . . . . . . . . 11 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑅 Cn (𝑅 ×t 𝑆)) = (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆))))
5044, 49eleqtrd 2828 . . . . . . . . . 10 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) ∈ (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆))))
51 cnima 23256 . . . . . . . . . 10 (((𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) ∈ (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → ((𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) “ 𝑥) ∈ 𝑅)
5250, 20, 51syl2anc 582 . . . . . . . . 9 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ((𝑡 𝑅 ↦ ⟨𝑡, (2nd𝑦)⟩) “ 𝑥) ∈ 𝑅)
539, 52eqeltrrid 2831 . . . . . . . 8 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∈ 𝑅)
54 opeq1 4873 . . . . . . . . . 10 (𝑡 = (1st𝑦) → ⟨𝑡, (2nd𝑦)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5554eleq1d 2811 . . . . . . . . 9 (𝑡 = (1st𝑦) → (⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥))
56 xp1st 8026 . . . . . . . . . 10 (𝑦 ∈ ( 𝑅 × 𝑆) → (1st𝑦) ∈ 𝑅)
5732, 56syl 17 . . . . . . . . 9 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (1st𝑦) ∈ 𝑅)
58 1st2nd2 8033 . . . . . . . . . . 11 (𝑦 ∈ ( 𝑅 × 𝑆) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
5932, 58syl 17 . . . . . . . . . 10 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
6059, 19eqeltrrd 2827 . . . . . . . . 9 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥)
6155, 57, 60elrabd 3684 . . . . . . . 8 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (1st𝑦) ∈ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥})
62 nlly2i 23467 . . . . . . . 8 ((𝑅 ∈ 𝑛-Locally Comp ∧ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∈ 𝑅 ∧ (1st𝑦) ∈ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥}) → ∃𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥}∃𝑢𝑅 ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))
637, 53, 61, 62syl3anc 1368 . . . . . . 7 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ∃𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥}∃𝑢𝑅 ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))
6410adantr 479 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑅 ∈ Top)
6516adantr 479 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑆 ∈ Top)
66 simprlr 778 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑢𝑅)
67 ssrab2 4075 . . . . . . . . . . . . . 14 {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ 𝑆
6867a1i 11 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ 𝑆)
69 incom 4201 . . . . . . . . . . . . . . . 16 ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) = (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})
70 simprll 777 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥})
7170elpwid 4608 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑠 ⊆ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥})
72 ssrab2 4075 . . . . . . . . . . . . . . . . . . . . . 22 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ⊆ 𝑅
7371, 72sstrdi 3993 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑠 𝑅)
7473adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑠 𝑅)
75 elpwi 4606 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝒫 𝑆𝑘 𝑆)
7675ad2antrl 726 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑘 𝑆)
77 eldif 3958 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡𝑥))
7877anbi1i 622 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ ((𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡𝑥) ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏))
79 anass 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡𝑥) ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ (¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏)))
8078, 79bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ (¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏)))
8180rexbii2 3080 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ↔ ∃𝑡 ∈ (𝑠 × 𝑘)(¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏))
82 ancom 459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ (((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ∧ ¬ 𝑡𝑥))
83 fveqeq2 6901 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = ⟨𝑎, 𝑢⟩ → (((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ↔ ((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏))
84 eleq1 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = ⟨𝑎, 𝑢⟩ → (𝑡𝑥 ↔ ⟨𝑎, 𝑢⟩ ∈ 𝑥))
8584notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = ⟨𝑎, 𝑢⟩ → (¬ 𝑡𝑥 ↔ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥))
8683, 85anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = ⟨𝑎, 𝑢⟩ → ((((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ∧ ¬ 𝑡𝑥) ↔ (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥)))
8782, 86bitrid 282 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = ⟨𝑎, 𝑢⟩ → ((¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥)))
8887rexxp 5841 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑡 ∈ (𝑠 × 𝑘)(¬ 𝑡𝑥 ∧ ((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏) ↔ ∃𝑎𝑠𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥))
8981, 88bitri 274 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ↔ ∃𝑎𝑠𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥))
90 simpl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑠 𝑅𝑘 𝑆) → 𝑠 𝑅)
9190sselda 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) → 𝑎 𝑅)
9291adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → 𝑎 𝑅)
93 simplr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) → 𝑘 𝑆)
9493sselda 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → 𝑢 𝑆)
9592, 94opelxpd 5713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → ⟨𝑎, 𝑢⟩ ∈ ( 𝑅 × 𝑆))
9695fvresd 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → ((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = (2nd ‘⟨𝑎, 𝑢⟩))
97 vex 3468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑎 ∈ V
98 vex 3468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑢 ∈ V
9997, 98op2nd 8003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (2nd ‘⟨𝑎, 𝑢⟩) = 𝑢
10096, 99eqtrdi 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → ((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑢)
101100eqeq1d 2728 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏𝑢 = 𝑏))
102101anbi1d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) ∧ 𝑢𝑘) → ((((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ (𝑢 = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥)))
103102rexbidva 3167 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) → (∃𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ ∃𝑢𝑘 (𝑢 = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥)))
104 opeq2 4874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = 𝑏 → ⟨𝑎, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
105104eleq1d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢 = 𝑏 → (⟨𝑎, 𝑢⟩ ∈ 𝑥 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
106105notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 = 𝑏 → (¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥 ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
107106ceqsrexbv 3642 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑢𝑘 (𝑢 = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ (𝑏𝑘 ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
108103, 107bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑎𝑠) → (∃𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ (𝑏𝑘 ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
109108rexbidva 3167 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 𝑅𝑘 𝑆) → (∃𝑎𝑠𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ ∃𝑎𝑠 (𝑏𝑘 ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
110 r19.42v 3181 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑎𝑠 (𝑏𝑘 ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥) ↔ (𝑏𝑘 ∧ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
111109, 110bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 𝑅𝑘 𝑆) → (∃𝑎𝑠𝑢𝑘 (((2nd ↾ ( 𝑅 × 𝑆))‘⟨𝑎, 𝑢⟩) = 𝑏 ∧ ¬ ⟨𝑎, 𝑢⟩ ∈ 𝑥) ↔ (𝑏𝑘 ∧ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
11289, 111bitrid 282 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 𝑅𝑘 𝑆) → (∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏 ↔ (𝑏𝑘 ∧ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
113 f2ndres 8019 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ↾ ( 𝑅 × 𝑆)):( 𝑅 × 𝑆)⟶ 𝑆
114 ffn 6719 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ↾ ( 𝑅 × 𝑆)):( 𝑅 × 𝑆)⟶ 𝑆 → (2nd ↾ ( 𝑅 × 𝑆)) Fn ( 𝑅 × 𝑆))
115113, 114ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ↾ ( 𝑅 × 𝑆)) Fn ( 𝑅 × 𝑆)
116 difss 4130 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘)
117 xpss12 5689 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 𝑅𝑘 𝑆) → (𝑠 × 𝑘) ⊆ ( 𝑅 × 𝑆))
118116, 117sstrid 3992 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 𝑅𝑘 𝑆) → ((𝑠 × 𝑘) ∖ 𝑥) ⊆ ( 𝑅 × 𝑆))
119 fvelimab 6966 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ↾ ( 𝑅 × 𝑆)) Fn ( 𝑅 × 𝑆) ∧ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ ( 𝑅 × 𝑆)) → (𝑏 ∈ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ ∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏))
120115, 118, 119sylancr 585 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 𝑅𝑘 𝑆) → (𝑏 ∈ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ ∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ ( 𝑅 × 𝑆))‘𝑡) = 𝑏))
121 eldif 3958 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏𝑘 ∧ ¬ 𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
122 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 𝑅𝑘 𝑆) → 𝑘 𝑆)
123122sselda 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑏𝑘) → 𝑏 𝑆)
124 sneq 4635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑣 = 𝑏 → {𝑣} = {𝑏})
125124xpeq2d 5704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑣 = 𝑏 → (𝑠 × {𝑣}) = (𝑠 × {𝑏}))
126125sseq1d 4012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 = 𝑏 → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ (𝑠 × {𝑏}) ⊆ 𝑥))
127 dfss3 3969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑠 × {𝑏}) ⊆ 𝑥 ↔ ∀𝑘 ∈ (𝑠 × {𝑏})𝑘𝑥)
128 eleq1 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = ⟨𝑎, 𝑡⟩ → (𝑘𝑥 ↔ ⟨𝑎, 𝑡⟩ ∈ 𝑥))
129128ralxp 5840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑘 ∈ (𝑠 × {𝑏})𝑘𝑥 ↔ ∀𝑎𝑠𝑡 ∈ {𝑏}⟨𝑎, 𝑡⟩ ∈ 𝑥)
130 vex 3468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑏 ∈ V
131 opeq2 4874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑏 → ⟨𝑎, 𝑡⟩ = ⟨𝑎, 𝑏⟩)
132131eleq1d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑏 → (⟨𝑎, 𝑡⟩ ∈ 𝑥 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
133130, 132ralsn 4682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑡 ∈ {𝑏}⟨𝑎, 𝑡⟩ ∈ 𝑥 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥)
134133ralbii 3083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑎𝑠𝑡 ∈ {𝑏}⟨𝑎, 𝑡⟩ ∈ 𝑥 ↔ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥)
135127, 129, 1343bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑠 × {𝑏}) ⊆ 𝑥 ↔ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥)
136126, 135bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 = 𝑏 → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥))
137136elrab3 3683 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 𝑆 → (𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥))
138123, 137syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑏𝑘) → (𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥))
139138notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑏𝑘) → (¬ 𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ¬ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥))
140 rexnal 3090 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥 ↔ ¬ ∀𝑎𝑠𝑎, 𝑏⟩ ∈ 𝑥)
141139, 140bitr4di 288 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑠 𝑅𝑘 𝑆) ∧ 𝑏𝑘) → (¬ 𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥))
142141pm5.32da 577 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 𝑅𝑘 𝑆) → ((𝑏𝑘 ∧ ¬ 𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏𝑘 ∧ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
143121, 142bitrid 282 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 𝑅𝑘 𝑆) → (𝑏 ∈ (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏𝑘 ∧ ∃𝑎𝑠 ¬ ⟨𝑎, 𝑏⟩ ∈ 𝑥)))
144112, 120, 1433bitr4d 310 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 𝑅𝑘 𝑆) → (𝑏 ∈ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ 𝑏 ∈ (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})))
145144eqrdv 2724 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 𝑅𝑘 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
14674, 76, 145syl2anc 582 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
147 difin 4262 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) = (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})
14865adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑆 ∈ Top)
14924restuni 23153 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ Top ∧ 𝑘 𝑆) → 𝑘 = (𝑆t 𝑘))
150148, 76, 149syl2anc 582 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑘 = (𝑆t 𝑘))
151150difeq1d 4119 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑘 ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) = ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})))
152147, 151eqtr3id 2780 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑘 ∖ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) = ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})))
153146, 152eqtrd 2766 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})))
15415ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑆 ∈ (ran 𝑘Gen ∩ Haus))
155154elin2d 4199 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑆 ∈ Haus)
156 df-ima 5687 . . . . . . . . . . . . . . . . . . . . . . 23 ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = ran ((2nd ↾ ( 𝑅 × 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥))
157 resres 5994 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ↾ ( 𝑅 × 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) = (2nd ↾ (( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)))
158 inss2 4230 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ((𝑠 × 𝑘) ∖ 𝑥)
159158, 116sstri 3990 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (𝑠 × 𝑘)
160 ssres2 6006 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (𝑠 × 𝑘) → (2nd ↾ (( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥))) ⊆ (2nd ↾ (𝑠 × 𝑘)))
161159, 160ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2nd ↾ (( 𝑅 × 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥))) ⊆ (2nd ↾ (𝑠 × 𝑘))
162157, 161eqsstri 4015 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ↾ ( 𝑅 × 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (2nd ↾ (𝑠 × 𝑘))
163162rnssi 5938 . . . . . . . . . . . . . . . . . . . . . . 23 ran ((2nd ↾ ( 𝑅 × 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ran (2nd ↾ (𝑠 × 𝑘))
164156, 163eqsstri 4015 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ran (2nd ↾ (𝑠 × 𝑘))
165 f2ndres 8019 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ↾ (𝑠 × 𝑘)):(𝑠 × 𝑘)⟶𝑘
166 frn 6726 . . . . . . . . . . . . . . . . . . . . . . 23 ((2nd ↾ (𝑠 × 𝑘)):(𝑠 × 𝑘)⟶𝑘 → ran (2nd ↾ (𝑠 × 𝑘)) ⊆ 𝑘)
167165, 166ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ran (2nd ↾ (𝑠 × 𝑘)) ⊆ 𝑘
168164, 167sstri 3990 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘
169168, 76sstrid 3992 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑆)
17012ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑅 ∈ (TopOn‘ 𝑅))
171148, 17sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑆 ∈ (TopOn‘ 𝑆))
172 tx2cn 23601 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
173170, 171, 172syl2anc 582 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
17427ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑅 ×t 𝑆) ∈ Top)
175116a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘))
176 vex 3468 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑠 ∈ V
177 vex 3468 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘 ∈ V
178176, 177xpex 7752 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 × 𝑘) ∈ V
179178a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑠 × 𝑘) ∈ V)
180 restabs 23156 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘) ∧ (𝑠 × 𝑘) ∈ V) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) = ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥)))
181174, 175, 179, 180syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) = ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥)))
18264adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑅 ∈ Top)
183154, 4syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑆 ∈ Top)
184176a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑠 ∈ V)
185 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑘 ∈ 𝒫 𝑆)
186 txrest 23622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑠 ∈ V ∧ 𝑘 ∈ 𝒫 𝑆)) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) = ((𝑅t 𝑠) ×t (𝑆t 𝑘)))
187182, 183, 184, 185, 186syl22anc 837 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) = ((𝑅t 𝑠) ×t (𝑆t 𝑘)))
188 simprr3 1220 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (𝑅t 𝑠) ∈ Comp)
189188adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑅t 𝑠) ∈ Comp)
190 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑆t 𝑘) ∈ Comp)
191 txcmp 23634 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅t 𝑠) ∈ Comp ∧ (𝑆t 𝑘) ∈ Comp) → ((𝑅t 𝑠) ×t (𝑆t 𝑘)) ∈ Comp)
192189, 190, 191syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑅t 𝑠) ×t (𝑆t 𝑘)) ∈ Comp)
193187, 192eqeltrd 2826 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Comp)
194 difin 4262 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 × 𝑘) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) = ((𝑠 × 𝑘) ∖ 𝑥)
19574, 76, 117syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑠 × 𝑘) ⊆ ( 𝑅 × 𝑆))
196182, 148, 25syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
197195, 196sseqtrd 4021 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑠 × 𝑘) ⊆ (𝑅 ×t 𝑆))
19828restuni 23153 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑠 × 𝑘) ⊆ (𝑅 ×t 𝑆)) → (𝑠 × 𝑘) = ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))
199174, 197, 198syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑠 × 𝑘) = ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))
200199difeq1d 4119 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑠 × 𝑘) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) = ( ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)))
201194, 200eqtr3id 2780 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑠 × 𝑘) ∖ 𝑥) = ( ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)))
202 resttop 23151 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑠 × 𝑘) ∈ V) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Top)
203174, 178, 202sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Top)
204 incom 4201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑠 × 𝑘) ∩ 𝑥) = (𝑥 ∩ (𝑠 × 𝑘))
20520ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆)))
206 kgeni 23528 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆)) ∧ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Comp) → (𝑥 ∩ (𝑠 × 𝑘)) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))
207205, 193, 206syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑥 ∩ (𝑠 × 𝑘)) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))
208204, 207eqeltrid 2830 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑠 × 𝑘) ∩ 𝑥) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))
209 eqid 2726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) = ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))
210209opncld 23024 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Top ∧ ((𝑠 × 𝑘) ∩ 𝑥) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) → ( ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))))
211203, 208, 210syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ( ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))))
212201, 211eqeltrd 2826 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑠 × 𝑘) ∖ 𝑥) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))))
213 cmpcld 23393 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Comp ∧ ((𝑠 × 𝑘) ∖ 𝑥) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp)
214193, 212, 213syl2anc 582 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp)
215181, 214eqeltrrd 2827 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp)
216 imacmp 23388 . . . . . . . . . . . . . . . . . . . . 21 (((2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ∧ ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp) → (𝑆t ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp)
217173, 215, 216syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑆t ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp)
21824hauscmp 23398 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ Haus ∧ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑆 ∧ (𝑆t ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆))
219155, 169, 217, 218syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆))
220168a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘)
22124restcldi 23164 . . . . . . . . . . . . . . . . . . 19 ((𝑘 𝑆 ∧ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆) ∧ ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘(𝑆t 𝑘)))
22276, 219, 220, 221syl3anc 1368 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((2nd ↾ ( 𝑅 × 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘(𝑆t 𝑘)))
223153, 222eqeltrrd 2827 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆t 𝑘)))
224 resttop 23151 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝒫 𝑆) → (𝑆t 𝑘) ∈ Top)
225148, 185, 224syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑆t 𝑘) ∈ Top)
226 inss1 4229 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑘
227226, 150sseqtrid 4033 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ (𝑆t 𝑘))
228 eqid 2726 . . . . . . . . . . . . . . . . . . 19 (𝑆t 𝑘) = (𝑆t 𝑘)
229228isopn2 23023 . . . . . . . . . . . . . . . . . 18 (((𝑆t 𝑘) ∈ Top ∧ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ (𝑆t 𝑘)) → ((𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆t 𝑘) ↔ ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆t 𝑘))))
230225, 227, 229syl2anc 582 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ((𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆t 𝑘) ↔ ( (𝑆t 𝑘) ∖ (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆t 𝑘))))
231223, 230mpbird 256 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → (𝑘 ∩ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆t 𝑘))
23269, 231eqeltrid 2830 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 𝑆 ∧ (𝑆t 𝑘) ∈ Comp)) → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆t 𝑘))
233232expr 455 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑘 ∈ 𝒫 𝑆) → ((𝑆t 𝑘) ∈ Comp → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆t 𝑘)))
234233ralrimiva 3136 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ∀𝑘 ∈ 𝒫 𝑆((𝑆t 𝑘) ∈ Comp → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆t 𝑘)))
23565, 17sylib 217 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑆 ∈ (TopOn‘ 𝑆))
236 elkgen 23527 . . . . . . . . . . . . . 14 (𝑆 ∈ (TopOn‘ 𝑆) → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆) ↔ ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ 𝑆 ∧ ∀𝑘 ∈ 𝒫 𝑆((𝑆t 𝑘) ∈ Comp → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆t 𝑘)))))
237235, 236syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆) ↔ ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ 𝑆 ∧ ∀𝑘 ∈ 𝒫 𝑆((𝑆t 𝑘) ∈ Comp → ({𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆t 𝑘)))))
23868, 234, 237mpbir2and 711 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆))
23915adantr 479 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑆 ∈ (ran 𝑘Gen ∩ Haus))
240239, 2syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑆 ∈ ran 𝑘Gen)
241 kgenidm 23538 . . . . . . . . . . . . 13 (𝑆 ∈ ran 𝑘Gen → (𝑘Gen‘𝑆) = 𝑆)
242240, 241syl 17 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (𝑘Gen‘𝑆) = 𝑆)
243238, 242eleqtrd 2828 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ 𝑆)
244 txopn 23593 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑢𝑅 ∧ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ 𝑆)) → (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆))
24564, 65, 66, 243, 244syl22anc 837 . . . . . . . . . 10 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆))
24659adantr 479 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
247 simprr1 1218 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (1st𝑦) ∈ 𝑢)
248 sneq 4635 . . . . . . . . . . . . . . 15 (𝑣 = (2nd𝑦) → {𝑣} = {(2nd𝑦)})
249248xpeq2d 5704 . . . . . . . . . . . . . 14 (𝑣 = (2nd𝑦) → (𝑠 × {𝑣}) = (𝑠 × {(2nd𝑦)}))
250249sseq1d 4012 . . . . . . . . . . . . 13 (𝑣 = (2nd𝑦) → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ (𝑠 × {(2nd𝑦)}) ⊆ 𝑥))
25134adantr 479 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (2nd𝑦) ∈ 𝑆)
252 relxp 5692 . . . . . . . . . . . . . . 15 Rel (𝑠 × {(2nd𝑦)})
253252a1i 11 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → Rel (𝑠 × {(2nd𝑦)}))
254 opelxp 5710 . . . . . . . . . . . . . . 15 (⟨𝑎, 𝑏⟩ ∈ (𝑠 × {(2nd𝑦)}) ↔ (𝑎𝑠𝑏 ∈ {(2nd𝑦)}))
25571sselda 3980 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑠) → 𝑎 ∈ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥})
256 opeq1 4873 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑎 → ⟨𝑡, (2nd𝑦)⟩ = ⟨𝑎, (2nd𝑦)⟩)
257256eleq1d 2811 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑎 → (⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥 ↔ ⟨𝑎, (2nd𝑦)⟩ ∈ 𝑥))
258257elrab 3682 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ↔ (𝑎 𝑅 ∧ ⟨𝑎, (2nd𝑦)⟩ ∈ 𝑥))
259258simprbi 495 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} → ⟨𝑎, (2nd𝑦)⟩ ∈ 𝑥)
260255, 259syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑠) → ⟨𝑎, (2nd𝑦)⟩ ∈ 𝑥)
261 elsni 4642 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ {(2nd𝑦)} → 𝑏 = (2nd𝑦))
262261opeq2d 4880 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ {(2nd𝑦)} → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd𝑦)⟩)
263262eleq1d 2811 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {(2nd𝑦)} → (⟨𝑎, 𝑏⟩ ∈ 𝑥 ↔ ⟨𝑎, (2nd𝑦)⟩ ∈ 𝑥))
264260, 263syl5ibrcom 246 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑠) → (𝑏 ∈ {(2nd𝑦)} → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
265264expimpd 452 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ((𝑎𝑠𝑏 ∈ {(2nd𝑦)}) → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
266254, 265biimtrid 241 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (⟨𝑎, 𝑏⟩ ∈ (𝑠 × {(2nd𝑦)}) → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
267253, 266relssdv 5786 . . . . . . . . . . . . 13 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (𝑠 × {(2nd𝑦)}) ⊆ 𝑥)
268250, 251, 267elrabd 3684 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (2nd𝑦) ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})
269247, 268opelxpd 5713 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
270246, 269eqeltrd 2826 . . . . . . . . . 10 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
271 relxp 5692 . . . . . . . . . . . 12 Rel (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})
272271a1i 11 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → Rel (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
273 opelxp 5710 . . . . . . . . . . . 12 (⟨𝑎, 𝑏⟩ ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑎𝑢𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))
274126elrab 3682 . . . . . . . . . . . . . . 15 (𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ (𝑏 𝑆 ∧ (𝑠 × {𝑏}) ⊆ 𝑥))
275274simprbi 495 . . . . . . . . . . . . . 14 (𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} → (𝑠 × {𝑏}) ⊆ 𝑥)
276 simprr2 1219 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → 𝑢𝑠)
277276sselda 3980 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑢) → 𝑎𝑠)
278 vsnid 4662 . . . . . . . . . . . . . . 15 𝑏 ∈ {𝑏}
279 opelxpi 5711 . . . . . . . . . . . . . . 15 ((𝑎𝑠𝑏 ∈ {𝑏}) → ⟨𝑎, 𝑏⟩ ∈ (𝑠 × {𝑏}))
280277, 278, 279sylancl 584 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑢) → ⟨𝑎, 𝑏⟩ ∈ (𝑠 × {𝑏}))
281 ssel 3974 . . . . . . . . . . . . . 14 ((𝑠 × {𝑏}) ⊆ 𝑥 → (⟨𝑎, 𝑏⟩ ∈ (𝑠 × {𝑏}) → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
282275, 280, 281syl2imc 41 . . . . . . . . . . . . 13 ((((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) ∧ 𝑎𝑢) → (𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
283282expimpd 452 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ((𝑎𝑢𝑏 ∈ {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
284273, 283biimtrid 241 . . . . . . . . . . 11 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (⟨𝑎, 𝑏⟩ ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → ⟨𝑎, 𝑏⟩ ∈ 𝑥))
285272, 284relssdv 5786 . . . . . . . . . 10 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥)
286 eleq2 2815 . . . . . . . . . . . 12 (𝑡 = (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → (𝑦𝑡𝑦 ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})))
287 sseq1 4006 . . . . . . . . . . . 12 (𝑡 = (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → (𝑡𝑥 ↔ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥))
288286, 287anbi12d 630 . . . . . . . . . . 11 (𝑡 = (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → ((𝑦𝑡𝑡𝑥) ↔ (𝑦 ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∧ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥)))
289288rspcev 3609 . . . . . . . . . 10 (((𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆) ∧ (𝑦 ∈ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∧ (𝑢 × {𝑣 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥)) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥))
290245, 270, 285, 289syl12anc 835 . . . . . . . . 9 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅) ∧ ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp))) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥))
291290expr 455 . . . . . . . 8 (((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) ∧ (𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥} ∧ 𝑢𝑅)) → (((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥)))
292291rexlimdvva 3202 . . . . . . 7 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → (∃𝑠 ∈ 𝒫 {𝑡 𝑅 ∣ ⟨𝑡, (2nd𝑦)⟩ ∈ 𝑥}∃𝑢𝑅 ((1st𝑦) ∈ 𝑢𝑢𝑠 ∧ (𝑅t 𝑠) ∈ Comp) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥)))
29363, 292mpd 15 . . . . . 6 ((((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦𝑥) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥))
294293ralrimiva 3136 . . . . 5 (((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → ∀𝑦𝑥𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥))
2956adantr 479 . . . . . 6 (((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → (𝑅 ×t 𝑆) ∈ Top)
296 eltop2 22965 . . . . . 6 ((𝑅 ×t 𝑆) ∈ Top → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥)))
297295, 296syl 17 . . . . 5 (((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑡 ∈ (𝑅 ×t 𝑆)(𝑦𝑡𝑡𝑥)))
298294, 297mpbird 256 . . . 4 (((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → 𝑥 ∈ (𝑅 ×t 𝑆))
299298ex 411 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆)) → 𝑥 ∈ (𝑅 ×t 𝑆)))
300299ssrdv 3986 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑘Gen‘(𝑅 ×t 𝑆)) ⊆ (𝑅 ×t 𝑆))
301 iskgen2 23539 . 2 ((𝑅 ×t 𝑆) ∈ ran 𝑘Gen ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑘Gen‘(𝑅 ×t 𝑆)) ⊆ (𝑅 ×t 𝑆)))
3026, 300, 301sylanbrc 581 1 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {crab 3420  Vcvv 3464  cdif 3945  cin 3947  wss 3948  𝒫 cpw 4599  {csn 4625  cop 4631   cuni 4907  cmpt 5228   I cid 5571   × cxp 5672  ccnv 5673  ran crn 5675  cres 5676  cima 5677  Rel wrel 5679   Fn wfn 6540  wf 6541  cfv 6545  (class class class)co 7415  1st c1st 7992  2nd c2nd 7993  t crest 17429  Topctop 22882  TopOnctopon 22899  Clsdccld 23007   Cn ccn 23215  Hauscha 23299  Compccmp 23377  𝑛-Locally cnlly 23456  𝑘Genckgen 23524   ×t ctx 23551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-int 4949  df-iun 4997  df-iin 4998  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7994  df-2nd 7995  df-1o 8487  df-2o 8488  df-map 8848  df-en 8966  df-dom 8967  df-fin 8969  df-fi 9446  df-rest 17431  df-topgen 17452  df-top 22883  df-topon 22900  df-bases 22936  df-cld 23010  df-ntr 23011  df-cls 23012  df-nei 23089  df-cn 23218  df-cnp 23219  df-haus 23306  df-cmp 23378  df-nlly 23458  df-kgen 23525  df-tx 23553
This theorem is referenced by: (None)
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