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Theorem xkococnlem 21672
Description: Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
xkococn.1 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
xkococn.s (𝜑𝑆 ∈ 𝑛-Locally Comp)
xkococn.k (𝜑𝐾 𝑅)
xkococn.c (𝜑 → (𝑅t 𝐾) ∈ Comp)
xkococn.v (𝜑𝑉𝑇)
xkococn.a (𝜑𝐴 ∈ (𝑆 Cn 𝑇))
xkococn.b (𝜑𝐵 ∈ (𝑅 Cn 𝑆))
xkococn.i (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)
Assertion
Ref Expression
xkococnlem (𝜑 → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑓,𝑔,,𝑧,𝑅   𝑆,𝑓,𝑔,𝑧   ,𝐾,𝑧   𝑇,𝑓,𝑔,,𝑧   𝑧,𝐹   ,𝑉,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑓,𝑔,)   𝐴(𝑓,𝑔,)   𝐵(𝑓,𝑔,)   𝑆()   𝐹(𝑓,𝑔,)   𝐾(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem xkococnlem
Dummy variables 𝑘 𝑎 𝑠 𝑢 𝑣 𝑤 𝑥 𝑦 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkococn.b . . . 4 (𝜑𝐵 ∈ (𝑅 Cn 𝑆))
2 xkococn.c . . . 4 (𝜑 → (𝑅t 𝐾) ∈ Comp)
3 imacmp 21410 . . . 4 ((𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝐾) ∈ Comp) → (𝑆t (𝐵𝐾)) ∈ Comp)
41, 2, 3syl2anc 575 . . 3 (𝜑 → (𝑆t (𝐵𝐾)) ∈ Comp)
5 xkococn.s . . . . . . . . 9 (𝜑𝑆 ∈ 𝑛-Locally Comp)
65adantr 468 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑆 ∈ 𝑛-Locally Comp)
7 xkococn.a . . . . . . . . . 10 (𝜑𝐴 ∈ (𝑆 Cn 𝑇))
8 xkococn.v . . . . . . . . . 10 (𝜑𝑉𝑇)
9 cnima 21279 . . . . . . . . . 10 ((𝐴 ∈ (𝑆 Cn 𝑇) ∧ 𝑉𝑇) → (𝐴𝑉) ∈ 𝑆)
107, 8, 9syl2anc 575 . . . . . . . . 9 (𝜑 → (𝐴𝑉) ∈ 𝑆)
1110adantr 468 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → (𝐴𝑉) ∈ 𝑆)
12 imaco 5851 . . . . . . . . . . 11 ((𝐴𝐵) “ 𝐾) = (𝐴 “ (𝐵𝐾))
13 xkococn.i . . . . . . . . . . 11 (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)
1412, 13syl5eqssr 3844 . . . . . . . . . 10 (𝜑 → (𝐴 “ (𝐵𝐾)) ⊆ 𝑉)
15 eqid 2805 . . . . . . . . . . . . 13 𝑆 = 𝑆
16 eqid 2805 . . . . . . . . . . . . 13 𝑇 = 𝑇
1715, 16cnf 21260 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝐴: 𝑆 𝑇)
18 ffun 6256 . . . . . . . . . . . 12 (𝐴: 𝑆 𝑇 → Fun 𝐴)
197, 17, 183syl 18 . . . . . . . . . . 11 (𝜑 → Fun 𝐴)
20 imassrn 5684 . . . . . . . . . . . . 13 (𝐵𝐾) ⊆ ran 𝐵
21 eqid 2805 . . . . . . . . . . . . . . 15 𝑅 = 𝑅
2221, 15cnf 21260 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝐵: 𝑅 𝑆)
23 frn 6259 . . . . . . . . . . . . . 14 (𝐵: 𝑅 𝑆 → ran 𝐵 𝑆)
241, 22, 233syl 18 . . . . . . . . . . . . 13 (𝜑 → ran 𝐵 𝑆)
2520, 24syl5ss 3806 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ⊆ 𝑆)
26 fdm 6261 . . . . . . . . . . . . 13 (𝐴: 𝑆 𝑇 → dom 𝐴 = 𝑆)
277, 17, 263syl 18 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝑆)
2825, 27sseqtr4d 3836 . . . . . . . . . . 11 (𝜑 → (𝐵𝐾) ⊆ dom 𝐴)
29 funimass3 6552 . . . . . . . . . . 11 ((Fun 𝐴 ∧ (𝐵𝐾) ⊆ dom 𝐴) → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3019, 28, 29syl2anc 575 . . . . . . . . . 10 (𝜑 → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3114, 30mpbid 223 . . . . . . . . 9 (𝜑 → (𝐵𝐾) ⊆ (𝐴𝑉))
3231sselda 3795 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑥 ∈ (𝐴𝑉))
33 nlly2i 21489 . . . . . . . 8 ((𝑆 ∈ 𝑛-Locally Comp ∧ (𝐴𝑉) ∈ 𝑆𝑥 ∈ (𝐴𝑉)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
346, 11, 32, 33syl3anc 1483 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
35 nllytop 21486 . . . . . . . . . . . . 13 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
365, 35syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Top)
3736ad3antrrr 712 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑆 ∈ Top)
38 imaexg 7330 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑅 Cn 𝑆) → (𝐵𝐾) ∈ V)
391, 38syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ∈ V)
4039ad3antrrr 712 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐵𝐾) ∈ V)
41 simprl 778 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑆)
42 elrestr 16290 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ (𝐵𝐾) ∈ V ∧ 𝑢𝑆) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
4337, 40, 41, 42syl3anc 1483 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
44 simprr1 1280 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥𝑢)
45 simpllr 784 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝐵𝐾))
4644, 45elind 3994 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝑢 ∩ (𝐵𝐾)))
47 inss1 4026 . . . . . . . . . . . 12 (𝑢 ∩ (𝐵𝐾)) ⊆ 𝑢
48 elpwi 4358 . . . . . . . . . . . . . . 15 (𝑠 ∈ 𝒫 (𝐴𝑉) → 𝑠 ⊆ (𝐴𝑉))
4948ad2antlr 709 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 ⊆ (𝐴𝑉))
50 elssuni 4657 . . . . . . . . . . . . . . . 16 ((𝐴𝑉) ∈ 𝑆 → (𝐴𝑉) ⊆ 𝑆)
5110, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑉) ⊆ 𝑆)
5251ad3antrrr 712 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐴𝑉) ⊆ 𝑆)
5349, 52sstrd 3805 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 𝑆)
54 simprr2 1282 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑠)
5515ssntr 21072 . . . . . . . . . . . . 13 (((𝑆 ∈ Top ∧ 𝑠 𝑆) ∧ (𝑢𝑆𝑢𝑠)) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5637, 53, 41, 54, 55syl22anc 858 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5747, 56syl5ss 3806 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠))
58 simprr3 1284 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑆t 𝑠) ∈ Comp)
5957, 58jca 503 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))
60 eleq2 2873 . . . . . . . . . . . 12 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → (𝑥𝑦𝑥 ∈ (𝑢 ∩ (𝐵𝐾))))
61 sseq1 3820 . . . . . . . . . . . . 13 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ (𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠)))
6261anbi1d 617 . . . . . . . . . . . 12 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6360, 62anbi12d 618 . . . . . . . . . . 11 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ (𝑢 ∩ (𝐵𝐾)) ∧ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6463rspcev 3501 . . . . . . . . . 10 (((𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)) ∧ (𝑥 ∈ (𝑢 ∩ (𝐵𝐾)) ∧ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6543, 46, 59, 64syl12anc 856 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6665rexlimdvaa 3219 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) → (∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6766reximdva 3203 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵𝐾)) → (∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6834, 67mpd 15 . . . . . 6 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
69 rexcom 3286 . . . . . . 7 (∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆t (𝐵𝐾))∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
70 r19.42v 3279 . . . . . . . 8 (∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7170rexbii 3228 . . . . . . 7 (∃𝑦 ∈ (𝑆t (𝐵𝐾))∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7269, 71bitri 266 . . . . . 6 (∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7368, 72sylib 209 . . . . 5 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7473ralrimiva 3153 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐵𝐾)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7515restuni 21176 . . . . . 6 ((𝑆 ∈ Top ∧ (𝐵𝐾) ⊆ 𝑆) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7636, 25, 75syl2anc 575 . . . . 5 (𝜑 → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7776raleqdv 3332 . . . 4 (𝜑 → (∀𝑥 ∈ (𝐵𝐾)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ ∀𝑥 (𝑆t (𝐵𝐾))∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
7874, 77mpbid 223 . . 3 (𝜑 → ∀𝑥 (𝑆t (𝐵𝐾))∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
79 eqid 2805 . . . 4 (𝑆t (𝐵𝐾)) = (𝑆t (𝐵𝐾))
80 fveq2 6405 . . . . . 6 (𝑠 = (𝑘𝑦) → ((int‘𝑆)‘𝑠) = ((int‘𝑆)‘(𝑘𝑦)))
8180sseq2d 3827 . . . . 5 (𝑠 = (𝑘𝑦) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦))))
82 oveq2 6879 . . . . . 6 (𝑠 = (𝑘𝑦) → (𝑆t 𝑠) = (𝑆t (𝑘𝑦)))
8382eleq1d 2869 . . . . 5 (𝑠 = (𝑘𝑦) → ((𝑆t 𝑠) ∈ Comp ↔ (𝑆t (𝑘𝑦)) ∈ Comp))
8481, 83anbi12d 618 . . . 4 (𝑠 = (𝑘𝑦) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))
8579, 84cmpcovf 21404 . . 3 (((𝑆t (𝐵𝐾)) ∈ Comp ∧ ∀𝑥 (𝑆t (𝐵𝐾))∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))) → ∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))))
864, 78, 85syl2anc 575 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))))
8776adantr 468 . . . . . . 7 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
8887eqeq1d 2807 . . . . . 6 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → ((𝐵𝐾) = 𝑤 (𝑆t (𝐵𝐾)) = 𝑤))
8988biimpar 465 . . . . 5 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝑆t (𝐵𝐾)) = 𝑤) → (𝐵𝐾) = 𝑤)
9036ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑆 ∈ Top)
91 cntop2 21255 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
927, 91syl 17 . . . . . . . . . . 11 (𝜑𝑇 ∈ Top)
9392ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑇 ∈ Top)
94 xkotop 21601 . . . . . . . . . 10 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
9590, 93, 94syl2anc 575 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑇 ^ko 𝑆) ∈ Top)
96 cntop1 21254 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
971, 96syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ Top)
9897ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑅 ∈ Top)
99 xkotop 21601 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
10098, 90, 99syl2anc 575 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆 ^ko 𝑅) ∈ Top)
101 simprrl 790 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘:𝑤⟶𝒫 (𝐴𝑉))
102101frnd 6260 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ 𝒫 (𝐴𝑉))
103 sspwuni 4799 . . . . . . . . . . . 12 (ran 𝑘 ⊆ 𝒫 (𝐴𝑉) ↔ ran 𝑘 ⊆ (𝐴𝑉))
104102, 103sylib 209 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ (𝐴𝑉))
10510ad2antrr 708 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ∈ 𝑆)
106105, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ⊆ 𝑆)
107104, 106sstrd 3805 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 𝑆)
108 ffn 6253 . . . . . . . . . . . . 13 (𝑘:𝑤⟶𝒫 (𝐴𝑉) → 𝑘 Fn 𝑤)
109 fniunfv 6726 . . . . . . . . . . . . 13 (𝑘 Fn 𝑤 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
110101, 108, 1093syl 18 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
111110oveq2d 6887 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t 𝑦𝑤 (𝑘𝑦)) = (𝑆t ran 𝑘))
112 inss2 4027 . . . . . . . . . . . . 13 (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin) ⊆ Fin
113 simplr 776 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin))
114112, 113sseldi 3793 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑤 ∈ Fin)
115 simprrr 791 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))
116 simpr 473 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t (𝑘𝑦)) ∈ Comp)
117116ralimi 3139 . . . . . . . . . . . . 13 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp)
118115, 117syl 17 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp)
11915fiuncmp 21417 . . . . . . . . . . . 12 ((𝑆 ∈ Top ∧ 𝑤 ∈ Fin ∧ ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
12090, 114, 118, 119syl3anc 1483 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
121111, 120eqeltrrd 2885 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t ran 𝑘) ∈ Comp)
1228ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑉𝑇)
12315, 90, 93, 107, 121, 122xkoopn 21602 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∈ (𝑇 ^ko 𝑆))
124 xkococn.k . . . . . . . . . . 11 (𝜑𝐾 𝑅)
125124ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐾 𝑅)
1262ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑅t 𝐾) ∈ Comp)
127110, 107eqsstrd 3833 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
128 iunss 4749 . . . . . . . . . . . . 13 ( 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
129127, 128sylib 209 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
13015ntropn 21063 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
131130ex 399 . . . . . . . . . . . . 13 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
132131ralimdv 3150 . . . . . . . . . . . 12 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
13390, 129, 132sylc 65 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
134 iunopn 20912 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
13590, 133, 134syl2anc 575 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
13621, 98, 90, 125, 126, 135xkoopn 21602 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ∈ (𝑆 ^ko 𝑅))
137 txopn 21615 . . . . . . . . 9 ((((𝑇 ^ko 𝑆) ∈ Top ∧ (𝑆 ^ko 𝑅) ∈ Top) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∈ (𝑇 ^ko 𝑆) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ∈ (𝑆 ^ko 𝑅))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)))
13895, 100, 123, 136, 137syl22anc 858 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)))
1397ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐴 ∈ (𝑆 Cn 𝑇))
140 imaiun 6724 . . . . . . . . . . . 12 (𝐴 𝑦𝑤 (𝑘𝑦)) = 𝑦𝑤 (𝐴 “ (𝑘𝑦))
141110imaeq2d 5673 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴 𝑦𝑤 (𝑘𝑦)) = (𝐴 ran 𝑘))
142140, 141syl5eqr 2853 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝐴 “ (𝑘𝑦)) = (𝐴 ran 𝑘))
143110, 104eqsstrd 3833 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
14419ad2antrr 708 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → Fun 𝐴)
145101, 108syl 17 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘 Fn 𝑤)
14627ad2antrr 708 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → dom 𝐴 = 𝑆)
147107, 146sseqtr4d 3836 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ dom 𝐴)
148 simpl1 1235 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → Fun 𝐴)
1491093ad2ant2 1157 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
150 simp3 1161 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ran 𝑘 ⊆ dom 𝐴)
151149, 150eqsstrd 3833 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
152 iunss 4749 . . . . . . . . . . . . . . . . . 18 ( 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
153151, 152sylib 209 . . . . . . . . . . . . . . . . 17 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
154153r19.21bi 3119 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → (𝑘𝑦) ⊆ dom 𝐴)
155 funimass3 6552 . . . . . . . . . . . . . . . 16 ((Fun 𝐴 ∧ (𝑘𝑦) ⊆ dom 𝐴) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
156148, 154, 155syl2anc 575 . . . . . . . . . . . . . . 15 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
157156ralbidva 3172 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → (∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
158 iunss 4749 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
159 iunss 4749 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉) ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
160157, 158, 1593bitr4g 305 . . . . . . . . . . . . 13 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
161144, 145, 147, 160syl3anc 1483 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
162143, 161mpbird 248 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
163142, 162eqsstr3d 3834 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴 ran 𝑘) ⊆ 𝑉)
164 imaeq1 5668 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑎 ran 𝑘) = (𝐴 ran 𝑘))
165164sseq1d 3826 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝐴 ran 𝑘) ⊆ 𝑉))
166165elrab 3558 . . . . . . . . . 10 (𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ↔ (𝐴 ∈ (𝑆 Cn 𝑇) ∧ (𝐴 ran 𝑘) ⊆ 𝑉))
167139, 163, 166sylanbrc 574 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉})
1681ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐵 ∈ (𝑅 Cn 𝑆))
169 simprl 778 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑤)
170 uniiun 4761 . . . . . . . . . . . 12 𝑤 = 𝑦𝑤 𝑦
171169, 170syl6eq 2855 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑦𝑤 𝑦)
172 simpl 470 . . . . . . . . . . . . 13 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
173172ralimi 3139 . . . . . . . . . . . 12 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
174 ss2iun 4724 . . . . . . . . . . . 12 (∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) → 𝑦𝑤 𝑦 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
175115, 173, 1743syl 18 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 𝑦 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
176171, 175eqsstrd 3833 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
177 imaeq1 5668 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝑏𝐾) = (𝐵𝐾))
178177sseq1d 3826 . . . . . . . . . . 11 (𝑏 = 𝐵 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
179178elrab 3558 . . . . . . . . . 10 (𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ↔ (𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
180168, 176, 179sylanbrc 574 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})
181 opelxpi 5345 . . . . . . . . 9 ((𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∧ 𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) → ⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}))
182167, 180, 181syl2anc 575 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}))
183 imaeq1 5668 . . . . . . . . . . . . . . 15 (𝑎 = 𝑢 → (𝑎 ran 𝑘) = (𝑢 ran 𝑘))
184183sseq1d 3826 . . . . . . . . . . . . . 14 (𝑎 = 𝑢 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝑢 ran 𝑘) ⊆ 𝑉))
185184elrab 3558 . . . . . . . . . . . . 13 (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ↔ (𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉))
186 imaeq1 5668 . . . . . . . . . . . . . . 15 (𝑏 = 𝑣 → (𝑏𝐾) = (𝑣𝐾))
187186sseq1d 3826 . . . . . . . . . . . . . 14 (𝑏 = 𝑣 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
188187elrab 3558 . . . . . . . . . . . . 13 (𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ↔ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
189185, 188anbi12i 614 . . . . . . . . . . . 12 ((𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ↔ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))))
190 simprll 788 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → 𝑢 ∈ (𝑆 Cn 𝑇))
191 simprrl 790 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → 𝑣 ∈ (𝑅 Cn 𝑆))
192 coeq1 5478 . . . . . . . . . . . . . . 15 (𝑓 = 𝑢 → (𝑓𝑔) = (𝑢𝑔))
193 coeq2 5479 . . . . . . . . . . . . . . 15 (𝑔 = 𝑣 → (𝑢𝑔) = (𝑢𝑣))
194 xkococn.1 . . . . . . . . . . . . . . 15 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
195 vex 3393 . . . . . . . . . . . . . . . 16 𝑢 ∈ V
196 vex 3393 . . . . . . . . . . . . . . . 16 𝑣 ∈ V
197195, 196coex 7345 . . . . . . . . . . . . . . 15 (𝑢𝑣) ∈ V
198192, 193, 194, 197ovmpt2 7023 . . . . . . . . . . . . . 14 ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ (𝑅 Cn 𝑆)) → (𝑢𝐹𝑣) = (𝑢𝑣))
199190, 191, 198syl2anc 575 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢𝐹𝑣) = (𝑢𝑣))
200 cnco 21280 . . . . . . . . . . . . . . 15 ((𝑣 ∈ (𝑅 Cn 𝑆) ∧ 𝑢 ∈ (𝑆 Cn 𝑇)) → (𝑢𝑣) ∈ (𝑅 Cn 𝑇))
201191, 190, 200syl2anc 575 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢𝑣) ∈ (𝑅 Cn 𝑇))
202 imaco 5851 . . . . . . . . . . . . . . 15 ((𝑢𝑣) “ 𝐾) = (𝑢 “ (𝑣𝐾))
203 simprrr 791 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
20415ntrss2 21071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦))
205204ex 399 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
206205ralimdv 3150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
20790, 129, 206sylc 65 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦))
208 ss2iun 4724 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ 𝑦𝑤 (𝑘𝑦))
209207, 208syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ 𝑦𝑤 (𝑘𝑦))
210209, 110sseqtrd 3835 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ ran 𝑘)
211210adantr 468 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ ran 𝑘)
212203, 211sstrd 3805 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑣𝐾) ⊆ ran 𝑘)
213 imass2 5708 . . . . . . . . . . . . . . . . 17 ((𝑣𝐾) ⊆ ran 𝑘 → (𝑢 “ (𝑣𝐾)) ⊆ (𝑢 ran 𝑘))
214212, 213syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢 “ (𝑣𝐾)) ⊆ (𝑢 ran 𝑘))
215 simprlr 789 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢 ran 𝑘) ⊆ 𝑉)
216214, 215sstrd 3805 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢 “ (𝑣𝐾)) ⊆ 𝑉)
217202, 216syl5eqss 3843 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → ((𝑢𝑣) “ 𝐾) ⊆ 𝑉)
218 imaeq1 5668 . . . . . . . . . . . . . . . 16 ( = (𝑢𝑣) → (𝐾) = ((𝑢𝑣) “ 𝐾))
219218sseq1d 3826 . . . . . . . . . . . . . . 15 ( = (𝑢𝑣) → ((𝐾) ⊆ 𝑉 ↔ ((𝑢𝑣) “ 𝐾) ⊆ 𝑉))
220219elrab 3558 . . . . . . . . . . . . . 14 ((𝑢𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ((𝑢𝑣) ∈ (𝑅 Cn 𝑇) ∧ ((𝑢𝑣) “ 𝐾) ⊆ 𝑉))
221201, 217, 220sylanbrc 574 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
222199, 221eqeltrd 2884 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))) → (𝑢𝐹𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
223189, 222sylan2b 583 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) ∧ (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) → (𝑢𝐹𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
224223ralrimivva 3158 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} (𝑢𝐹𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
225194mpt2fun 6989 . . . . . . . . . . 11 Fun 𝐹
226 ssrab2 3881 . . . . . . . . . . . . 13 {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇)
227 ssrab2 3881 . . . . . . . . . . . . 13 {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ⊆ (𝑅 Cn 𝑆)
228 xpss12 5323 . . . . . . . . . . . . 13 (({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ⊆ (𝑅 Cn 𝑆)) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
229226, 227, 228mp2an 675 . . . . . . . . . . . 12 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
230 vex 3393 . . . . . . . . . . . . . 14 𝑓 ∈ V
231 vex 3393 . . . . . . . . . . . . . 14 𝑔 ∈ V
232230, 231coex 7345 . . . . . . . . . . . . 13 (𝑓𝑔) ∈ V
233194, 232dmmpt2 7470 . . . . . . . . . . . 12 dom 𝐹 = ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
234229, 233sseqtr4i 3832 . . . . . . . . . . 11 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ dom 𝐹
235 funimassov 7038 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) ⊆ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} (𝑢𝐹𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))
236225, 234, 235mp2an 675 . . . . . . . . . 10 ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) ⊆ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} (𝑢𝐹𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
237224, 236sylibr 225 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) ⊆ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})
238 funimass3 6552 . . . . . . . . . 10 ((Fun 𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) ⊆ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
239225, 234, 238mp2an 675 . . . . . . . . 9 ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})) ⊆ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))
240237, 239sylib 209 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))
241 eleq2 2873 . . . . . . . . . 10 (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) → (⟨𝐴, 𝐵⟩ ∈ 𝑧 ↔ ⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))})))
242 sseq1 3820 . . . . . . . . . 10 (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) → (𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}) ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
243241, 242anbi12d 618 . . . . . . . . 9 (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) → ((⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})) ↔ (⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
244243rspcev 3501 . . . . . . . 8 ((({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∧ (⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
245138, 182, 240, 244syl12anc 856 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
246245expr 446 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝐵𝐾) = 𝑤) → ((𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
247246exlimdv 2026 . . . . 5 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝐵𝐾) = 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
24889, 247syldan 581 . . . 4 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝑆t (𝐵𝐾)) = 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
249248expimpd 443 . . 3 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → (( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
250249rexlimdva 3218 . 2 (𝜑 → (∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
25186, 250mpd 15 1 (𝜑 → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2158  wral 3095  wrex 3096  {crab 3099  Vcvv 3390  cin 3765  wss 3766  𝒫 cpw 4348  cop 4373   cuni 4626   ciun 4708   × cxp 5306  ccnv 5307  dom cdm 5308  ran crn 5309  cima 5311  ccom 5312  Fun wfun 6092   Fn wfn 6093  wf 6094  cfv 6098  (class class class)co 6871  cmpt2 6873  Fincfn 8189  t crest 16282  Topctop 20907  intcnt 21031   Cn ccn 21238  Compccmp 21399  𝑛-Locally cnlly 21478   ×t ctx 21573   ^ko cxko 21574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-map 8091  df-en 8190  df-dom 8191  df-fin 8193  df-fi 8553  df-rest 16284  df-topgen 16305  df-top 20908  df-topon 20925  df-bases 20960  df-ntr 21034  df-nei 21112  df-cn 21241  df-cmp 21400  df-nlly 21480  df-tx 21575  df-xko 21576
This theorem is referenced by:  xkococn  21673
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