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Theorem txtube 22224
 Description: The "tube lemma". If 𝑋 is compact and there is an open set 𝑈 containing the line 𝑋 × {𝐴}, then there is a "tube" 𝑋 × 𝑢 for some neighborhood 𝑢 of 𝐴 which is entirely contained within 𝑈. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
txtube.x 𝑋 = 𝑅
txtube.y 𝑌 = 𝑆
txtube.r (𝜑𝑅 ∈ Comp)
txtube.s (𝜑𝑆 ∈ Top)
txtube.w (𝜑𝑈 ∈ (𝑅 ×t 𝑆))
txtube.u (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
txtube.a (𝜑𝐴𝑌)
Assertion
Ref Expression
txtube (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑆   𝑢,𝑌   𝜑,𝑢   𝑢,𝑈   𝑢,𝑋

Proof of Theorem txtube
Dummy variables 𝑡 𝑓 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txtube.r . . 3 (𝜑𝑅 ∈ Comp)
2 eleq1 2899 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑢 × 𝑣) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣)))
32anbi1d 632 . . . . . . 7 (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
432rexbidv 3286 . . . . . 6 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
5 txtube.w . . . . . . . 8 (𝜑𝑈 ∈ (𝑅 ×t 𝑆))
6 txtube.s . . . . . . . . 9 (𝜑𝑆 ∈ Top)
7 eltx 22152 . . . . . . . . 9 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Top) → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
81, 6, 7syl2anc 587 . . . . . . . 8 (𝜑 → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
95, 8mpbid 235 . . . . . . 7 (𝜑 → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
109adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
11 txtube.u . . . . . . . 8 (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
1211adantr 484 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13 id 22 . . . . . . . 8 (𝑥𝑋𝑥𝑋)
14 txtube.a . . . . . . . . 9 (𝜑𝐴𝑌)
15 snidg 4572 . . . . . . . . 9 (𝐴𝑌𝐴 ∈ {𝐴})
1614, 15syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
17 opelxpi 5565 . . . . . . . 8 ((𝑥𝑋𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1813, 16, 17syl2anr 599 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1912, 18sseldd 3944 . . . . . 6 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
204, 10, 19rspcdva 3602 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21 opelxp 5564 . . . . . . . . . 10 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥𝑢𝐴𝑣))
2221anbi1i 626 . . . . . . . . 9 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23 anass 472 . . . . . . . . 9 (((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2422, 23bitri 278 . . . . . . . 8 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2524rexbii 3235 . . . . . . 7 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26 r19.42v 3335 . . . . . . 7 (∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2725, 26bitri 278 . . . . . 6 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2827rexbii 3235 . . . . 5 (∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2920, 28sylib 221 . . . 4 ((𝜑𝑥𝑋) → ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
3029ralrimiva 3170 . . 3 (𝜑 → ∀𝑥𝑋𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31 txtube.x . . . 4 𝑋 = 𝑅
32 eleq2 2900 . . . . 5 (𝑣 = (𝑓𝑢) → (𝐴𝑣𝐴 ∈ (𝑓𝑢)))
33 xpeq2 5549 . . . . . 6 (𝑣 = (𝑓𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓𝑢)))
3433sseq1d 3974 . . . . 5 (𝑣 = (𝑓𝑢) → ((𝑢 × 𝑣) ⊆ 𝑈 ↔ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))
3532, 34anbi12d 633 . . . 4 (𝑣 = (𝑓𝑢) → ((𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))
3631, 35cmpcovf 21975 . . 3 ((𝑅 ∈ Comp ∧ ∀𝑥𝑋𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈))) → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))))
371, 30, 36syl2anc 587 . 2 (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))))
38 rint0 4889 . . . . . . . . . 10 (ran 𝑓 = ∅ → (𝑌 ran 𝑓) = 𝑌)
3938adantl 485 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ran 𝑓) = 𝑌)
40 txtube.y . . . . . . . . . . . 12 𝑌 = 𝑆
4140topopn 21490 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑌𝑆)
426, 41syl 17 . . . . . . . . . 10 (𝜑𝑌𝑆)
4342ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → 𝑌𝑆)
4439, 43eqeltrd 2912 . . . . . . . 8 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ran 𝑓) ∈ 𝑆)
456ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → 𝑆 ∈ Top)
46 simprrl 780 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡𝑆)
4746frnd 6494 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ran 𝑓𝑆)
4847adantr 484 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
49 simpr 488 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ≠ ∅)
50 simplr 768 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
5150elin2d 4151 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ Fin)
5246ffnd 6488 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓 Fn 𝑡)
53 dffn4 6569 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
5452, 53sylib 221 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡onto→ran 𝑓)
55 fofi 8786 . . . . . . . . . . . . . . 15 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓 ∈ Fin)
5651, 54, 55syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ran 𝑓 ∈ Fin)
5756adantr 484 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ∈ Fin)
58 fiinopn 21485 . . . . . . . . . . . . . 14 (𝑆 ∈ Top → ((ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ran 𝑓𝑆))
5958imp 410 . . . . . . . . . . . . 13 ((𝑆 ∈ Top ∧ (ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ran 𝑓𝑆)
6045, 48, 49, 57, 59syl13anc 1369 . . . . . . . . . . . 12 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
61 elssuni 4841 . . . . . . . . . . . 12 ( ran 𝑓𝑆 ran 𝑓 𝑆)
6260, 61syl 17 . . . . . . . . . . 11 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 𝑆)
6362, 40sseqtrrdi 3994 . . . . . . . . . 10 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑌)
64 sseqin2 4167 . . . . . . . . . 10 ( ran 𝑓𝑌 ↔ (𝑌 ran 𝑓) = ran 𝑓)
6563, 64sylib 221 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) = ran 𝑓)
6665, 60eqeltrd 2912 . . . . . . . 8 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) ∈ 𝑆)
6744, 66pm2.61dane 3094 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑌 ran 𝑓) ∈ 𝑆)
6814ad2antrr 725 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴𝑌)
69 simprrr 781 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))
70 simpl 486 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓𝑢))
7170ralimi 3148 . . . . . . . . . . 11 (∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢))
7269, 71syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢))
73 eliin 4897 . . . . . . . . . . 11 (𝐴𝑌 → (𝐴 𝑢𝑡 (𝑓𝑢) ↔ ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢)))
7468, 73syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝐴 𝑢𝑡 (𝑓𝑢) ↔ ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢)))
7572, 74mpbird 260 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 𝑢𝑡 (𝑓𝑢))
76 fniinfv 6715 . . . . . . . . . 10 (𝑓 Fn 𝑡 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
7752, 76syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
7875, 77eleqtrd 2914 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 ran 𝑓)
7968, 78elind 4146 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ (𝑌 ran 𝑓))
80 simprl 770 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑋 = 𝑡)
81 uniiun 4955 . . . . . . . . . . 11 𝑡 = 𝑢𝑡 𝑢
8280, 81syl6eq 2872 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑋 = 𝑢𝑡 𝑢)
8382xpeq1d 5557 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) = ( 𝑢𝑡 𝑢 × (𝑌 ran 𝑓)))
84 xpiundir 5596 . . . . . . . . 9 ( 𝑢𝑡 𝑢 × (𝑌 ran 𝑓)) = 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓))
8583, 84syl6eq 2872 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) = 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)))
86 simpr 488 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
8786ralimi 3148 . . . . . . . . . . 11 (∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → ∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
8869, 87syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
89 inss2 4181 . . . . . . . . . . . . 13 (𝑌 ran 𝑓) ⊆ ran 𝑓
9076adantr 484 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
91 iinss2 4954 . . . . . . . . . . . . . . 15 (𝑢𝑡 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9291adantl 485 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9390, 92eqsstrrd 3982 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑡𝑢𝑡) → ran 𝑓 ⊆ (𝑓𝑢))
9489, 93sstrid 3954 . . . . . . . . . . . 12 ((𝑓 Fn 𝑡𝑢𝑡) → (𝑌 ran 𝑓) ⊆ (𝑓𝑢))
95 xpss2 5548 . . . . . . . . . . . 12 ((𝑌 ran 𝑓) ⊆ (𝑓𝑢) → (𝑢 × (𝑌 ran 𝑓)) ⊆ (𝑢 × (𝑓𝑢)))
96 sstr2 3950 . . . . . . . . . . . 12 ((𝑢 × (𝑌 ran 𝑓)) ⊆ (𝑢 × (𝑓𝑢)) → ((𝑢 × (𝑓𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
9794, 95, 963syl 18 . . . . . . . . . . 11 ((𝑓 Fn 𝑡𝑢𝑡) → ((𝑢 × (𝑓𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
9897ralimdva 3165 . . . . . . . . . 10 (𝑓 Fn 𝑡 → (∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈 → ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
9952, 88, 98sylc 65 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
100 iunss 4942 . . . . . . . . 9 ( 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈 ↔ ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
10199, 100sylibr 237 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
10285, 101eqsstrd 3981 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)
103 eleq2 2900 . . . . . . . . 9 (𝑢 = (𝑌 ran 𝑓) → (𝐴𝑢𝐴 ∈ (𝑌 ran 𝑓)))
104 xpeq2 5549 . . . . . . . . . 10 (𝑢 = (𝑌 ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ran 𝑓)))
105104sseq1d 3974 . . . . . . . . 9 (𝑢 = (𝑌 ran 𝑓) → ((𝑋 × 𝑢) ⊆ 𝑈 ↔ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈))
106103, 105anbi12d 633 . . . . . . . 8 (𝑢 = (𝑌 ran 𝑓) → ((𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑌 ran 𝑓) ∧ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)))
107106rspcev 3600 . . . . . . 7 (((𝑌 ran 𝑓) ∈ 𝑆 ∧ (𝐴 ∈ (𝑌 ran 𝑓) ∧ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
10867, 79, 102, 107syl12anc 835 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
109108expr 460 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
110109exlimdv 1935 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
111110expimpd 457 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
112111rexlimdva 3270 . 2 (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
11337, 112mpd 15 1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3007  ∀wral 3126  ∃wrex 3127   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266  𝒫 cpw 4512  {csn 4540  ⟨cop 4546  ∪ cuni 4811  ∩ cint 4849  ∪ ciun 4892  ∩ ciin 4893   × cxp 5526  ran crn 5529   Fn wfn 6323  ⟶wf 6324  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7130  Fincfn 8484  Topctop 21477  Compccmp 21970   ×t ctx 22144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-fin 8488  df-topgen 16696  df-top 21478  df-cmp 21971  df-tx 22146 This theorem is referenced by:  txcmplem1  22225  xkoinjcn  22271  cvmlift2lem12  32569
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