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Theorem xkococnlem 21684
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 21422 . . . 4 ((𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝐾) ∈ Comp) → (𝑆t (𝐵𝐾)) ∈ Comp)
41, 2, 3syl2anc 696 . . 3 (𝜑 → (𝑆t (𝐵𝐾)) ∈ Comp)
5 xkococn.s . . . . . . . . 9 (𝜑𝑆 ∈ 𝑛-Locally Comp)
65adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑆 ∈ 𝑛-Locally Comp)
7 xkococn.a . . . . . . . . . 10 (𝜑𝐴 ∈ (𝑆 Cn 𝑇))
8 xkococn.v . . . . . . . . . 10 (𝜑𝑉𝑇)
9 cnima 21291 . . . . . . . . . 10 ((𝐴 ∈ (𝑆 Cn 𝑇) ∧ 𝑉𝑇) → (𝐴𝑉) ∈ 𝑆)
107, 8, 9syl2anc 696 . . . . . . . . 9 (𝜑 → (𝐴𝑉) ∈ 𝑆)
1110adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → (𝐴𝑉) ∈ 𝑆)
12 imaco 5801 . . . . . . . . . . 11 ((𝐴𝐵) “ 𝐾) = (𝐴 “ (𝐵𝐾))
13 xkococn.i . . . . . . . . . . 11 (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)
1412, 13syl5eqssr 3791 . . . . . . . . . 10 (𝜑 → (𝐴 “ (𝐵𝐾)) ⊆ 𝑉)
15 eqid 2760 . . . . . . . . . . . . 13 𝑆 = 𝑆
16 eqid 2760 . . . . . . . . . . . . 13 𝑇 = 𝑇
1715, 16cnf 21272 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝐴: 𝑆 𝑇)
18 ffun 6209 . . . . . . . . . . . 12 (𝐴: 𝑆 𝑇 → Fun 𝐴)
197, 17, 183syl 18 . . . . . . . . . . 11 (𝜑 → Fun 𝐴)
20 imassrn 5635 . . . . . . . . . . . . 13 (𝐵𝐾) ⊆ ran 𝐵
21 eqid 2760 . . . . . . . . . . . . . . 15 𝑅 = 𝑅
2221, 15cnf 21272 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝐵: 𝑅 𝑆)
23 frn 6214 . . . . . . . . . . . . . 14 (𝐵: 𝑅 𝑆 → ran 𝐵 𝑆)
241, 22, 233syl 18 . . . . . . . . . . . . 13 (𝜑 → ran 𝐵 𝑆)
2520, 24syl5ss 3755 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ⊆ 𝑆)
26 fdm 6212 . . . . . . . . . . . . 13 (𝐴: 𝑆 𝑇 → dom 𝐴 = 𝑆)
277, 17, 263syl 18 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝑆)
2825, 27sseqtr4d 3783 . . . . . . . . . . 11 (𝜑 → (𝐵𝐾) ⊆ dom 𝐴)
29 funimass3 6497 . . . . . . . . . . 11 ((Fun 𝐴 ∧ (𝐵𝐾) ⊆ dom 𝐴) → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3019, 28, 29syl2anc 696 . . . . . . . . . 10 (𝜑 → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3114, 30mpbid 222 . . . . . . . . 9 (𝜑 → (𝐵𝐾) ⊆ (𝐴𝑉))
3231sselda 3744 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑥 ∈ (𝐴𝑉))
33 nlly2i 21501 . . . . . . . 8 ((𝑆 ∈ 𝑛-Locally Comp ∧ (𝐴𝑉) ∈ 𝑆𝑥 ∈ (𝐴𝑉)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
346, 11, 32, 33syl3anc 1477 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
35 nllytop 21498 . . . . . . . . . . . . 13 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
365, 35syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Top)
3736ad3antrrr 768 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑆 ∈ Top)
38 imaexg 7269 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑅 Cn 𝑆) → (𝐵𝐾) ∈ V)
391, 38syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ∈ V)
4039ad3antrrr 768 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐵𝐾) ∈ V)
41 simprl 811 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑆)
42 elrestr 16311 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ (𝐵𝐾) ∈ V ∧ 𝑢𝑆) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
4337, 40, 41, 42syl3anc 1477 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
44 simprr1 1273 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥𝑢)
45 simpllr 817 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝐵𝐾))
4644, 45elind 3941 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝑢 ∩ (𝐵𝐾)))
47 inss1 3976 . . . . . . . . . . . 12 (𝑢 ∩ (𝐵𝐾)) ⊆ 𝑢
48 elpwi 4312 . . . . . . . . . . . . . . 15 (𝑠 ∈ 𝒫 (𝐴𝑉) → 𝑠 ⊆ (𝐴𝑉))
4948ad2antlr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 ⊆ (𝐴𝑉))
50 elssuni 4619 . . . . . . . . . . . . . . . 16 ((𝐴𝑉) ∈ 𝑆 → (𝐴𝑉) ⊆ 𝑆)
5110, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑉) ⊆ 𝑆)
5251ad3antrrr 768 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐴𝑉) ⊆ 𝑆)
5349, 52sstrd 3754 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 𝑆)
54 simprr2 1275 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑠)
5515ssntr 21084 . . . . . . . . . . . . 13 (((𝑆 ∈ Top ∧ 𝑠 𝑆) ∧ (𝑢𝑆𝑢𝑠)) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5637, 53, 41, 54, 55syl22anc 1478 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5747, 56syl5ss 3755 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠))
58 simprr3 1277 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑆t 𝑠) ∈ Comp)
5957, 58jca 555 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))
60 eleq2 2828 . . . . . . . . . . . 12 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → (𝑥𝑦𝑥 ∈ (𝑢 ∩ (𝐵𝐾))))
61 sseq1 3767 . . . . . . . . . . . . 13 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ (𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠)))
6261anbi1d 743 . . . . . . . . . . . 12 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6360, 62anbi12d 749 . . . . . . . . . . 11 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ (𝑢 ∩ (𝐵𝐾)) ∧ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6463rspcev 3449 . . . . . . . . . 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 1475 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6665rexlimdvaa 3170 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) → (∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6766reximdva 3155 . . . . . . 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 3237 . . . . . . 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 3230 . . . . . . . 8 (∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7170rexbii 3179 . . . . . . 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 264 . . . . . 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 208 . . . . 5 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7473ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐵𝐾)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7515restuni 21188 . . . . . 6 ((𝑆 ∈ Top ∧ (𝐵𝐾) ⊆ 𝑆) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7636, 25, 75syl2anc 696 . . . . 5 (𝜑 → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7776raleqdv 3283 . . . 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 222 . . 3 (𝜑 → ∀𝑥 (𝑆t (𝐵𝐾))∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
79 eqid 2760 . . . 4 (𝑆t (𝐵𝐾)) = (𝑆t (𝐵𝐾))
80 fveq2 6353 . . . . . 6 (𝑠 = (𝑘𝑦) → ((int‘𝑆)‘𝑠) = ((int‘𝑆)‘(𝑘𝑦)))
8180sseq2d 3774 . . . . 5 (𝑠 = (𝑘𝑦) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦))))
82 oveq2 6822 . . . . . 6 (𝑠 = (𝑘𝑦) → (𝑆t 𝑠) = (𝑆t (𝑘𝑦)))
8382eleq1d 2824 . . . . 5 (𝑠 = (𝑘𝑦) → ((𝑆t 𝑠) ∈ Comp ↔ (𝑆t (𝑘𝑦)) ∈ Comp))
8481, 83anbi12d 749 . . . 4 (𝑠 = (𝑘𝑦) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))
8579, 84cmpcovf 21416 . . 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 696 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))))
8776adantr 472 . . . . . . 7 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
8887eqeq1d 2762 . . . . . 6 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → ((𝐵𝐾) = 𝑤 (𝑆t (𝐵𝐾)) = 𝑤))
8988biimpar 503 . . . . 5 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝑆t (𝐵𝐾)) = 𝑤) → (𝐵𝐾) = 𝑤)
9036ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑆 ∈ Top)
91 cntop2 21267 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
927, 91syl 17 . . . . . . . . . . 11 (𝜑𝑇 ∈ Top)
9392ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑇 ∈ Top)
94 xkotop 21613 . . . . . . . . . 10 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
9590, 93, 94syl2anc 696 . . . . . . . . 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 21266 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
971, 96syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ Top)
9897ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑅 ∈ Top)
99 xkotop 21613 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
10098, 90, 99syl2anc 696 . . . . . . . . 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 823 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘:𝑤⟶𝒫 (𝐴𝑉))
102 frn 6214 . . . . . . . . . . . . 13 (𝑘:𝑤⟶𝒫 (𝐴𝑉) → ran 𝑘 ⊆ 𝒫 (𝐴𝑉))
103101, 102syl 17 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ 𝒫 (𝐴𝑉))
104 sspwuni 4763 . . . . . . . . . . . 12 (ran 𝑘 ⊆ 𝒫 (𝐴𝑉) ↔ ran 𝑘 ⊆ (𝐴𝑉))
105103, 104sylib 208 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ (𝐴𝑉))
10610ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ∈ 𝑆)
107106, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ⊆ 𝑆)
108105, 107sstrd 3754 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 𝑆)
109 ffn 6206 . . . . . . . . . . . . 13 (𝑘:𝑤⟶𝒫 (𝐴𝑉) → 𝑘 Fn 𝑤)
110 fniunfv 6669 . . . . . . . . . . . . 13 (𝑘 Fn 𝑤 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
111101, 109, 1103syl 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)
112111oveq2d 6830 . . . . . . . . . . 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))
113 inss2 3977 . . . . . . . . . . . . 13 (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin) ⊆ Fin
114 simplr 809 . . . . . . . . . . . . 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))
115113, 114sseldi 3742 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑤 ∈ Fin)
116 simprrr 824 . . . . . . . . . . . . 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))
117 simpr 479 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t (𝑘𝑦)) ∈ Comp)
118117ralimi 3090 . . . . . . . . . . . . 13 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp)
119116, 118syl 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)
12015fiuncmp 21429 . . . . . . . . . . . 12 ((𝑆 ∈ Top ∧ 𝑤 ∈ Fin ∧ ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
12190, 115, 119, 120syl3anc 1477 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
122112, 121eqeltrrd 2840 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t ran 𝑘) ∈ Comp)
1238ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑉𝑇)
12415, 90, 93, 108, 122, 123xkoopn 21614 . . . . . . . . 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))
125 xkococn.k . . . . . . . . . . 11 (𝜑𝐾 𝑅)
126125ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐾 𝑅)
1272ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑅t 𝐾) ∈ Comp)
128111, 108eqsstrd 3780 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
129 iunss 4713 . . . . . . . . . . . . 13 ( 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
130128, 129sylib 208 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
13115ntropn 21075 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
132131ex 449 . . . . . . . . . . . . 13 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
133132ralimdv 3101 . . . . . . . . . . . 12 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
13490, 130, 133sylc 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)
135 iunopn 20925 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
13690, 134, 135syl2anc 696 . . . . . . . . . 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)
13721, 98, 90, 126, 127, 136xkoopn 21614 . . . . . . . . 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))
138 txopn 21627 . . . . . . . . 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)))
13995, 100, 124, 137, 138syl22anc 1478 . . . . . . . 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)))
1407ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐴 ∈ (𝑆 Cn 𝑇))
141 imaiun 6667 . . . . . . . . . . . 12 (𝐴 𝑦𝑤 (𝑘𝑦)) = 𝑦𝑤 (𝐴 “ (𝑘𝑦))
142111imaeq2d 5624 . . . . . . . . . . . 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))
143141, 142syl5eqr 2808 . . . . . . . . . . 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))
144111, 105eqsstrd 3780 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
14519ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → Fun 𝐴)
146101, 109syl 17 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘 Fn 𝑤)
14727ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → dom 𝐴 = 𝑆)
148108, 147sseqtr4d 3783 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ dom 𝐴)
149 simpl1 1228 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → Fun 𝐴)
1501103ad2ant2 1129 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
151 simp3 1133 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ran 𝑘 ⊆ dom 𝐴)
152150, 151eqsstrd 3780 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
153 iunss 4713 . . . . . . . . . . . . . . . . . 18 ( 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
154152, 153sylib 208 . . . . . . . . . . . . . . . . 17 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
155154r19.21bi 3070 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → (𝑘𝑦) ⊆ dom 𝐴)
156 funimass3 6497 . . . . . . . . . . . . . . . 16 ((Fun 𝐴 ∧ (𝑘𝑦) ⊆ dom 𝐴) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
157149, 155, 156syl2anc 696 . . . . . . . . . . . . . . 15 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
158157ralbidva 3123 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → (∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
159 iunss 4713 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
160 iunss 4713 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉) ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
161158, 159, 1603bitr4g 303 . . . . . . . . . . . . 13 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
162145, 146, 148, 161syl3anc 1477 . . . . . . . . . . . 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)))
163144, 162mpbird 247 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
164143, 163eqsstr3d 3781 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴 ran 𝑘) ⊆ 𝑉)
165 imaeq1 5619 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑎 ran 𝑘) = (𝐴 ran 𝑘))
166165sseq1d 3773 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝐴 ran 𝑘) ⊆ 𝑉))
167166elrab 3504 . . . . . . . . . 10 (𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ↔ (𝐴 ∈ (𝑆 Cn 𝑇) ∧ (𝐴 ran 𝑘) ⊆ 𝑉))
168140, 164, 167sylanbrc 701 . . . . . . . . 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})
1691ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐵 ∈ (𝑅 Cn 𝑆))
170 simprl 811 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑤)
171 uniiun 4725 . . . . . . . . . . . 12 𝑤 = 𝑦𝑤 𝑦
172170, 171syl6eq 2810 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑦𝑤 𝑦)
173 simpl 474 . . . . . . . . . . . . 13 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
174173ralimi 3090 . . . . . . . . . . . 12 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
175 ss2iun 4688 . . . . . . . . . . . 12 (∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) → 𝑦𝑤 𝑦 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
176116, 174, 1753syl 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)))
177172, 176eqsstrd 3780 . . . . . . . . . 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)))
178 imaeq1 5619 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝑏𝐾) = (𝐵𝐾))
179178sseq1d 3773 . . . . . . . . . . 11 (𝑏 = 𝐵 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
180179elrab 3504 . . . . . . . . . 10 (𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ↔ (𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
181169, 177, 180sylanbrc 701 . . . . . . . . 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))})
182 opelxpi 5305 . . . . . . . . 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))}))
183168, 181, 182syl2anc 696 . . . . . . . 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))}))
184 imaeq1 5619 . . . . . . . . . . . . . . 15 (𝑎 = 𝑢 → (𝑎 ran 𝑘) = (𝑢 ran 𝑘))
185184sseq1d 3773 . . . . . . . . . . . . . 14 (𝑎 = 𝑢 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝑢 ran 𝑘) ⊆ 𝑉))
186185elrab 3504 . . . . . . . . . . . . 13 (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ↔ (𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉))
187 imaeq1 5619 . . . . . . . . . . . . . . 15 (𝑏 = 𝑣 → (𝑏𝐾) = (𝑣𝐾))
188187sseq1d 3773 . . . . . . . . . . . . . 14 (𝑏 = 𝑣 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
189188elrab 3504 . . . . . . . . . . . . 13 (𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ↔ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
190186, 189anbi12i 735 . . . . . . . . . . . 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)))))
191 simprll 821 . . . . . . . . . . . . . 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))
192 simprrl 823 . . . . . . . . . . . . . 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))
193 coeq1 5435 . . . . . . . . . . . . . . 15 (𝑓 = 𝑢 → (𝑓𝑔) = (𝑢𝑔))
194 coeq2 5436 . . . . . . . . . . . . . . 15 (𝑔 = 𝑣 → (𝑢𝑔) = (𝑢𝑣))
195 xkococn.1 . . . . . . . . . . . . . . 15 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
196 vex 3343 . . . . . . . . . . . . . . . 16 𝑢 ∈ V
197 vex 3343 . . . . . . . . . . . . . . . 16 𝑣 ∈ V
198196, 197coex 7284 . . . . . . . . . . . . . . 15 (𝑢𝑣) ∈ V
199193, 194, 195, 198ovmpt2 6962 . . . . . . . . . . . . . 14 ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ (𝑅 Cn 𝑆)) → (𝑢𝐹𝑣) = (𝑢𝑣))
200191, 192, 199syl2anc 696 . . . . . . . . . . . . 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))
201 cnco 21292 . . . . . . . . . . . . . . 15 ((𝑣 ∈ (𝑅 Cn 𝑆) ∧ 𝑢 ∈ (𝑆 Cn 𝑇)) → (𝑢𝑣) ∈ (𝑅 Cn 𝑇))
202192, 191, 201syl2anc 696 . . . . . . . . . . . . . 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))
203 imaco 5801 . . . . . . . . . . . . . . 15 ((𝑢𝑣) “ 𝐾) = (𝑢 “ (𝑣𝐾))
204 simprrr 824 . . . . . . . . . . . . . . . . . 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)))
20515ntrss2 21083 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦))
206205ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
207206ralimdv 3101 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
20890, 130, 207sylc 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))
209 ss2iun 4688 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ 𝑦𝑤 (𝑘𝑦))
210208, 209syl 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))
211210, 111sseqtrd 3782 . . . . . . . . . . . . . . . . . . 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)
212211adantr 472 . . . . . . . . . . . . . . . . . 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)
213204, 212sstrd 3754 . . . . . . . . . . . . . . . . 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)
214 imass2 5659 . . . . . . . . . . . . . . . . 17 ((𝑣𝐾) ⊆ ran 𝑘 → (𝑢 “ (𝑣𝐾)) ⊆ (𝑢 ran 𝑘))
215213, 214syl 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))
216 simprlr 822 . . . . . . . . . . . . . . . 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)
217215, 216sstrd 3754 . . . . . . . . . . . . . . 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)
218203, 217syl5eqss 3790 . . . . . . . . . . . . . 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)
219 imaeq1 5619 . . . . . . . . . . . . . . . 16 ( = (𝑢𝑣) → (𝐾) = ((𝑢𝑣) “ 𝐾))
220219sseq1d 3773 . . . . . . . . . . . . . . 15 ( = (𝑢𝑣) → ((𝐾) ⊆ 𝑉 ↔ ((𝑢𝑣) “ 𝐾) ⊆ 𝑉))
221220elrab 3504 . . . . . . . . . . . . . 14 ((𝑢𝑣) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉} ↔ ((𝑢𝑣) ∈ (𝑅 Cn 𝑇) ∧ ((𝑢𝑣) “ 𝐾) ⊆ 𝑉))
222202, 218, 221sylanbrc 701 . . . . . . . . . . . . 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})
223200, 222eqeltrd 2839 . . . . . . . . . . . 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})
224190, 223sylan2b 493 . . . . . . . . . . 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})
225224ralrimivva 3109 . . . . . . . . . 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})
226195mpt2fun 6928 . . . . . . . . . . 11 Fun 𝐹
227 ssrab2 3828 . . . . . . . . . . . . 13 {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇)
228 ssrab2 3828 . . . . . . . . . . . . 13 {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ⊆ (𝑅 Cn 𝑆)
229 xpss12 5281 . . . . . . . . . . . . 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)))
230227, 228, 229mp2an 710 . . . . . . . . . . . 12 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
231 vex 3343 . . . . . . . . . . . . . 14 𝑓 ∈ V
232 vex 3343 . . . . . . . . . . . . . 14 𝑔 ∈ V
233231, 232coex 7284 . . . . . . . . . . . . 13 (𝑓𝑔) ∈ V
234195, 233dmmpt2 7409 . . . . . . . . . . . 12 dom 𝐹 = ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
235230, 234sseqtr4i 3779 . . . . . . . . . . 11 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ dom 𝐹
236 funimassov 6977 . . . . . . . . . . 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}))
237226, 235, 236mp2an 710 . . . . . . . . . 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})
238225, 237sylibr 224 . . . . . . . . 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})
239 funimass3 6497 . . . . . . . . . 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})))
240226, 235, 239mp2an 710 . . . . . . . . 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}))
241238, 240sylib 208 . . . . . . . 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}))
242 eleq2 2828 . . . . . . . . . 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))})))
243 sseq1 3767 . . . . . . . . . 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})))
244242, 243anbi12d 749 . . . . . . . . 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}))))
245244rspcev 3449 . . . . . . . 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})))
246139, 183, 241, 245syl12anc 1475 . . . . . . 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})))
247246expr 644 . . . . . 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}))))
248247exlimdv 2010 . . . . 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}))))
24989, 248syldan 488 . . . 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}))))
250249expimpd 630 . . 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}))))
251250rexlimdva 3169 . 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}))))
25286, 251mpd 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 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cin 3714  wss 3715  𝒫 cpw 4302  cop 4327   cuni 4588   ciun 4672   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  Fincfn 8123  t crest 16303  Topctop 20920  intcnt 21043   Cn ccn 21250  Compccmp 21411  𝑛-Locally cnlly 21490   ×t ctx 21585   ^ko cxko 21586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-fin 8127  df-fi 8484  df-rest 16305  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972  df-ntr 21046  df-nei 21124  df-cn 21253  df-cmp 21412  df-nlly 21492  df-tx 21587  df-xko 21588
This theorem is referenced by:  xkococn  21685
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