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Theorem xkococnlem 23683
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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 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 23421 . . . 4 ((𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝐾) ∈ Comp) → (𝑆t (𝐵𝐾)) ∈ Comp)
41, 2, 3syl2anc 584 . . 3 (𝜑 → (𝑆t (𝐵𝐾)) ∈ Comp)
5 xkococn.s . . . . . . . . 9 (𝜑𝑆 ∈ 𝑛-Locally Comp)
65adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑆 ∈ 𝑛-Locally Comp)
7 xkococn.a . . . . . . . . . 10 (𝜑𝐴 ∈ (𝑆 Cn 𝑇))
8 xkococn.v . . . . . . . . . 10 (𝜑𝑉𝑇)
9 cnima 23289 . . . . . . . . . 10 ((𝐴 ∈ (𝑆 Cn 𝑇) ∧ 𝑉𝑇) → (𝐴𝑉) ∈ 𝑆)
107, 8, 9syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐴𝑉) ∈ 𝑆)
1110adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → (𝐴𝑉) ∈ 𝑆)
12 imaco 6273 . . . . . . . . . . 11 ((𝐴𝐵) “ 𝐾) = (𝐴 “ (𝐵𝐾))
13 xkococn.i . . . . . . . . . . 11 (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)
1412, 13eqsstrrid 4045 . . . . . . . . . 10 (𝜑 → (𝐴 “ (𝐵𝐾)) ⊆ 𝑉)
15 eqid 2735 . . . . . . . . . . . . 13 𝑆 = 𝑆
16 eqid 2735 . . . . . . . . . . . . 13 𝑇 = 𝑇
1715, 16cnf 23270 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝐴: 𝑆 𝑇)
18 ffun 6740 . . . . . . . . . . . 12 (𝐴: 𝑆 𝑇 → Fun 𝐴)
197, 17, 183syl 18 . . . . . . . . . . 11 (𝜑 → Fun 𝐴)
20 imassrn 6091 . . . . . . . . . . . . 13 (𝐵𝐾) ⊆ ran 𝐵
21 eqid 2735 . . . . . . . . . . . . . . 15 𝑅 = 𝑅
2221, 15cnf 23270 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝐵: 𝑅 𝑆)
23 frn 6744 . . . . . . . . . . . . . 14 (𝐵: 𝑅 𝑆 → ran 𝐵 𝑆)
241, 22, 233syl 18 . . . . . . . . . . . . 13 (𝜑 → ran 𝐵 𝑆)
2520, 24sstrid 4007 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ⊆ 𝑆)
26 fdm 6746 . . . . . . . . . . . . 13 (𝐴: 𝑆 𝑇 → dom 𝐴 = 𝑆)
277, 17, 263syl 18 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝑆)
2825, 27sseqtrrd 4037 . . . . . . . . . . 11 (𝜑 → (𝐵𝐾) ⊆ dom 𝐴)
29 funimass3 7074 . . . . . . . . . . 11 ((Fun 𝐴 ∧ (𝐵𝐾) ⊆ dom 𝐴) → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3019, 28, 29syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐴 “ (𝐵𝐾)) ⊆ 𝑉 ↔ (𝐵𝐾) ⊆ (𝐴𝑉)))
3114, 30mpbid 232 . . . . . . . . 9 (𝜑 → (𝐵𝐾) ⊆ (𝐴𝑉))
3231sselda 3995 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵𝐾)) → 𝑥 ∈ (𝐴𝑉))
33 nlly2i 23500 . . . . . . . 8 ((𝑆 ∈ 𝑛-Locally Comp ∧ (𝐴𝑉) ∈ 𝑆𝑥 ∈ (𝐴𝑉)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
346, 11, 32, 33syl3anc 1370 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))
35 nllytop 23497 . . . . . . . . . . . . 13 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
365, 35syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Top)
3736ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑆 ∈ Top)
38 imaexg 7936 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑅 Cn 𝑆) → (𝐵𝐾) ∈ V)
391, 38syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐵𝐾) ∈ V)
4039ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐵𝐾) ∈ V)
41 simprl 771 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑆)
42 elrestr 17475 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ (𝐵𝐾) ∈ V ∧ 𝑢𝑆) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
4337, 40, 41, 42syl3anc 1370 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ∈ (𝑆t (𝐵𝐾)))
44 simprr1 1220 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥𝑢)
45 simpllr 776 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝐵𝐾))
4644, 45elind 4210 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝑢 ∩ (𝐵𝐾)))
47 inss1 4245 . . . . . . . . . . . 12 (𝑢 ∩ (𝐵𝐾)) ⊆ 𝑢
48 elpwi 4612 . . . . . . . . . . . . . . 15 (𝑠 ∈ 𝒫 (𝐴𝑉) → 𝑠 ⊆ (𝐴𝑉))
4948ad2antlr 727 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 ⊆ (𝐴𝑉))
50 elssuni 4942 . . . . . . . . . . . . . . . 16 ((𝐴𝑉) ∈ 𝑆 → (𝐴𝑉) ⊆ 𝑆)
5110, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑉) ⊆ 𝑆)
5251ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝐴𝑉) ⊆ 𝑆)
5349, 52sstrd 4006 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑠 𝑆)
54 simprr2 1221 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢𝑠)
5515ssntr 23082 . . . . . . . . . . . . 13 (((𝑆 ∈ Top ∧ 𝑠 𝑆) ∧ (𝑢𝑆𝑢𝑠)) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5637, 53, 41, 54, 55syl22anc 839 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
5747, 56sstrid 4007 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠))
58 simprr3 1222 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → (𝑆t 𝑠) ∈ Comp)
5957, 58jca 511 . . . . . . . . . 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 cleq1lem 15018 . . . . . . . . . . . 12 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6260, 61anbi12d 632 . . . . . . . . . . 11 (𝑦 = (𝑢 ∩ (𝐵𝐾)) → ((𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ (𝑢 ∩ (𝐵𝐾)) ∧ ((𝑢 ∩ (𝐵𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6362rspcev 3622 . . . . . . . . . 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)))
6443, 46, 59, 63syl12anc 837 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) ∧ (𝑢𝑆 ∧ (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
6564rexlimdvaa 3154 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐵𝐾)) ∧ 𝑠 ∈ 𝒫 (𝐴𝑉)) → (∃𝑢𝑆 (𝑥𝑢𝑢𝑠 ∧ (𝑆t 𝑠) ∈ Comp) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp))))
6665reximdva 3166 . . . . . . 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))))
6734, 66mpd 15 . . . . . 6 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑠 ∈ 𝒫 (𝐴𝑉)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
68 rexcom 3288 . . . . . . 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)))
69 r19.42v 3189 . . . . . . . 8 (∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑥𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)) ↔ (𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7069rexbii 3092 . . . . . . 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)))
7168, 70bitri 275 . . . . . 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)))
7267, 71sylib 218 . . . . 5 ((𝜑𝑥 ∈ (𝐵𝐾)) → ∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7372ralrimiva 3144 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐵𝐾)∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
7415restuni 23186 . . . . 5 ((𝑆 ∈ Top ∧ (𝐵𝐾) ⊆ 𝑆) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7536, 25, 74syl2anc 584 . . . 4 (𝜑 → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
7673, 75raleqtrdv 3326 . . 3 (𝜑 → ∀𝑥 (𝑆t (𝐵𝐾))∃𝑦 ∈ (𝑆t (𝐵𝐾))(𝑥𝑦 ∧ ∃𝑠 ∈ 𝒫 (𝐴𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp)))
77 eqid 2735 . . . 4 (𝑆t (𝐵𝐾)) = (𝑆t (𝐵𝐾))
78 fveq2 6907 . . . . . 6 (𝑠 = (𝑘𝑦) → ((int‘𝑆)‘𝑠) = ((int‘𝑆)‘(𝑘𝑦)))
7978sseq2d 4028 . . . . 5 (𝑠 = (𝑘𝑦) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦))))
80 oveq2 7439 . . . . . 6 (𝑠 = (𝑘𝑦) → (𝑆t 𝑠) = (𝑆t (𝑘𝑦)))
8180eleq1d 2824 . . . . 5 (𝑠 = (𝑘𝑦) → ((𝑆t 𝑠) ∈ Comp ↔ (𝑆t (𝑘𝑦)) ∈ Comp))
8279, 81anbi12d 632 . . . 4 (𝑠 = (𝑘𝑦) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆t 𝑠) ∈ Comp) ↔ (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))
8377, 82cmpcovf 23415 . . 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))))
844, 76, 83syl2anc 584 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))))
8575adantr 480 . . . . . . 7 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → (𝐵𝐾) = (𝑆t (𝐵𝐾)))
8685eqeq1d 2737 . . . . . 6 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → ((𝐵𝐾) = 𝑤 (𝑆t (𝐵𝐾)) = 𝑤))
8786biimpar 477 . . . . 5 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝑆t (𝐵𝐾)) = 𝑤) → (𝐵𝐾) = 𝑤)
8836ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑆 ∈ Top)
89 cntop2 23265 . . . . . . . . . . . 12 (𝐴 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
907, 89syl 17 . . . . . . . . . . 11 (𝜑𝑇 ∈ Top)
9190ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑇 ∈ Top)
92 xkotop 23612 . . . . . . . . . 10 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ Top)
9388, 91, 92syl2anc 584 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑇ko 𝑆) ∈ Top)
94 cntop1 23264 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
951, 94syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ Top)
9695ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑅 ∈ Top)
97 xkotop 23612 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
9896, 88, 97syl2anc 584 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆ko 𝑅) ∈ Top)
99 simprrl 781 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘:𝑤⟶𝒫 (𝐴𝑉))
10099frnd 6745 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ 𝒫 (𝐴𝑉))
101 sspwuni 5105 . . . . . . . . . . . 12 (ran 𝑘 ⊆ 𝒫 (𝐴𝑉) ↔ ran 𝑘 ⊆ (𝐴𝑉))
102100, 101sylib 218 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ (𝐴𝑉))
10310ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ∈ 𝑆)
104103, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴𝑉) ⊆ 𝑆)
105102, 104sstrd 4006 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 𝑆)
106 ffn 6737 . . . . . . . . . . . . 13 (𝑘:𝑤⟶𝒫 (𝐴𝑉) → 𝑘 Fn 𝑤)
107 fniunfv 7267 . . . . . . . . . . . . 13 (𝑘 Fn 𝑤 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
10899, 106, 1073syl 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)
109108oveq2d 7447 . . . . . . . . . . 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))
110 simplr 769 . . . . . . . . . . . . 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))
111110elin2d 4215 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑤 ∈ Fin)
112 simprrr 782 . . . . . . . . . . . . 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))
113 simpr 484 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t (𝑘𝑦)) ∈ Comp)
114113ralimi 3081 . . . . . . . . . . . . 13 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp)
115112, 114syl 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)
11615fiuncmp 23428 . . . . . . . . . . . 12 ((𝑆 ∈ Top ∧ 𝑤 ∈ Fin ∧ ∀𝑦𝑤 (𝑆t (𝑘𝑦)) ∈ Comp) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
11788, 111, 115, 116syl3anc 1370 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑆t 𝑦𝑤 (𝑘𝑦)) ∈ Comp)
118109, 117eqeltrrd 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)
1198ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑉𝑇)
12015, 88, 91, 105, 118, 119xkoopn 23613 . . . . . . . . 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} ∈ (𝑇ko 𝑆))
121 xkococn.k . . . . . . . . . . 11 (𝜑𝐾 𝑅)
122121ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐾 𝑅)
1232ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝑅t 𝐾) ∈ Comp)
124108, 105eqsstrd 4034 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
125 iunss 5050 . . . . . . . . . . . . 13 ( 𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
126124, 125sylib 218 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆)
12715ntropn 23073 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
128127ex 412 . . . . . . . . . . . . 13 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
129128ralimdv 3167 . . . . . . . . . . . 12 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆))
13088, 126, 129sylc 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)
131 iunopn 22920 . . . . . . . . . . 11 ((𝑆 ∈ Top ∧ ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ∈ 𝑆)
13288, 130, 131syl2anc 584 . . . . . . . . . 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)
13321, 96, 88, 122, 123, 132xkoopn 23613 . . . . . . . . 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))} ∈ (𝑆ko 𝑅))
134 txopn 23626 . . . . . . . . 9 ((((𝑇ko 𝑆) ∈ Top ∧ (𝑆ko 𝑅) ∈ Top) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ∈ (𝑇ko 𝑆) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ∈ (𝑆ko 𝑅))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)))
13593, 98, 120, 133, 134syl22anc 839 . . . . . . . 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))}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)))
136 imaeq1 6075 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 ran 𝑘) = (𝐴 ran 𝑘))
137136sseq1d 4027 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝐴 ran 𝑘) ⊆ 𝑉))
1387ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐴 ∈ (𝑆 Cn 𝑇))
139 imaiun 7265 . . . . . . . . . . . 12 (𝐴 𝑦𝑤 (𝑘𝑦)) = 𝑦𝑤 (𝐴 “ (𝑘𝑦))
140108imaeq2d 6080 . . . . . . . . . . . 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))
141139, 140eqtr3id 2789 . . . . . . . . . . 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))
142108, 102eqsstrd 4034 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
14319ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → Fun 𝐴)
14499, 106syl 17 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑘 Fn 𝑤)
14527ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → dom 𝐴 = 𝑆)
146105, 145sseqtrrd 4037 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ dom 𝐴)
147 simpl1 1190 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → Fun 𝐴)
1481073ad2ant2 1133 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) = ran 𝑘)
149 simp3 1137 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ran 𝑘 ⊆ dom 𝐴)
150148, 149eqsstrd 4034 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
151 iunss 5050 . . . . . . . . . . . . . . . . . 18 ( 𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
152150, 151sylib 218 . . . . . . . . . . . . . . . . 17 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ∀𝑦𝑤 (𝑘𝑦) ⊆ dom 𝐴)
153152r19.21bi 3249 . . . . . . . . . . . . . . . 16 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → (𝑘𝑦) ⊆ dom 𝐴)
154 funimass3 7074 . . . . . . . . . . . . . . . 16 ((Fun 𝐴 ∧ (𝑘𝑦) ⊆ dom 𝐴) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
155147, 153, 154syl2anc 584 . . . . . . . . . . . . . . 15 (((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦𝑤) → ((𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ (𝑘𝑦) ⊆ (𝐴𝑉)))
156155ralbidva 3174 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → (∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
157 iunss 5050 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 ↔ ∀𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
158 iunss 5050 . . . . . . . . . . . . . 14 ( 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉) ↔ ∀𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉))
159156, 157, 1583bitr4g 314 . . . . . . . . . . . . 13 ((Fun 𝐴𝑘 Fn 𝑤 ran 𝑘 ⊆ dom 𝐴) → ( 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉 𝑦𝑤 (𝑘𝑦) ⊆ (𝐴𝑉)))
160143, 144, 146, 159syl3anc 1370 . . . . . . . . . . . 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)))
161142, 160mpbird 257 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝑦𝑤 (𝐴 “ (𝑘𝑦)) ⊆ 𝑉)
162141, 161eqsstrrd 4035 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐴 ran 𝑘) ⊆ 𝑉)
163137, 138, 162elrabd 3697 . . . . . . . . 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})
164 imaeq1 6075 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏𝐾) = (𝐵𝐾))
165164sseq1d 4027 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝐵𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
1661ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → 𝐵 ∈ (𝑅 Cn 𝑆))
167 simprl 771 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑤)
168 uniiun 5063 . . . . . . . . . . . 12 𝑤 = 𝑦𝑤 𝑦
169167, 168eqtrdi 2791 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → (𝐵𝐾) = 𝑦𝑤 𝑦)
170 simpl 482 . . . . . . . . . . . . 13 ((𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
171170ralimi 3081 . . . . . . . . . . . 12 (∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp) → ∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)))
172 ss2iun 5015 . . . . . . . . . . . 12 (∀𝑦𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) → 𝑦𝑤 𝑦 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)))
173112, 171, 1723syl 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)))
174169, 173eqsstrd 4034 . . . . . . . . . 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)))
175165, 166, 174elrabd 3697 . . . . . . . . 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))})
176163, 175opelxpd 5728 . . . . . . . 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))}))
177 imaeq1 6075 . . . . . . . . . . . . . . 15 (𝑎 = 𝑢 → (𝑎 ran 𝑘) = (𝑢 ran 𝑘))
178177sseq1d 4027 . . . . . . . . . . . . . 14 (𝑎 = 𝑢 → ((𝑎 ran 𝑘) ⊆ 𝑉 ↔ (𝑢 ran 𝑘) ⊆ 𝑉))
179178elrab 3695 . . . . . . . . . . . . 13 (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ↔ (𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 ran 𝑘) ⊆ 𝑉))
180 imaeq1 6075 . . . . . . . . . . . . . . 15 (𝑏 = 𝑣 → (𝑏𝐾) = (𝑣𝐾))
181180sseq1d 4027 . . . . . . . . . . . . . 14 (𝑏 = 𝑣 → ((𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ↔ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
182181elrab 3695 . . . . . . . . . . . . 13 (𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ↔ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))))
183179, 182anbi12i 628 . . . . . . . . . . . 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)))))
184 simprll 779 . . . . . . . . . . . . . 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))
185 simprrl 781 . . . . . . . . . . . . . 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))
186 coeq1 5871 . . . . . . . . . . . . . . 15 (𝑓 = 𝑢 → (𝑓𝑔) = (𝑢𝑔))
187 coeq2 5872 . . . . . . . . . . . . . . 15 (𝑔 = 𝑣 → (𝑢𝑔) = (𝑢𝑣))
188 xkococn.1 . . . . . . . . . . . . . . 15 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
189 vex 3482 . . . . . . . . . . . . . . . 16 𝑢 ∈ V
190 vex 3482 . . . . . . . . . . . . . . . 16 𝑣 ∈ V
191189, 190coex 7953 . . . . . . . . . . . . . . 15 (𝑢𝑣) ∈ V
192186, 187, 188, 191ovmpo 7593 . . . . . . . . . . . . . 14 ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ (𝑅 Cn 𝑆)) → (𝑢𝐹𝑣) = (𝑢𝑣))
193184, 185, 192syl2anc 584 . . . . . . . . . . . . 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))
194 imaeq1 6075 . . . . . . . . . . . . . . 15 ( = (𝑢𝑣) → (𝐾) = ((𝑢𝑣) “ 𝐾))
195194sseq1d 4027 . . . . . . . . . . . . . 14 ( = (𝑢𝑣) → ((𝐾) ⊆ 𝑉 ↔ ((𝑢𝑣) “ 𝐾) ⊆ 𝑉))
196 cnco 23290 . . . . . . . . . . . . . . 15 ((𝑣 ∈ (𝑅 Cn 𝑆) ∧ 𝑢 ∈ (𝑆 Cn 𝑇)) → (𝑢𝑣) ∈ (𝑅 Cn 𝑇))
197185, 184, 196syl2anc 584 . . . . . . . . . . . . . 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))
198 imaco 6273 . . . . . . . . . . . . . . 15 ((𝑢𝑣) “ 𝐾) = (𝑢 “ (𝑣𝐾))
199 simprrr 782 . . . . . . . . . . . . . . . . . 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)))
20015ntrss2 23081 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ Top ∧ (𝑘𝑦) ⊆ 𝑆) → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦))
201200ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ Top → ((𝑘𝑦) ⊆ 𝑆 → ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
202201ralimdv 3167 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ∈ Top → (∀𝑦𝑤 (𝑘𝑦) ⊆ 𝑆 → ∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦)))
20388, 126, 202sylc 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))
204 ss2iun 5015 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ (𝑘𝑦) → 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦)) ⊆ 𝑦𝑤 (𝑘𝑦))
205203, 204syl 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))
206205, 108sseqtrd 4036 . . . . . . . . . . . . . . . . . . 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)
207206adantr 480 . . . . . . . . . . . . . . . . . 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)
208199, 207sstrd 4006 . . . . . . . . . . . . . . . . 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)
209 imass2 6123 . . . . . . . . . . . . . . . . 17 ((𝑣𝐾) ⊆ ran 𝑘 → (𝑢 “ (𝑣𝐾)) ⊆ (𝑢 ran 𝑘))
210208, 209syl 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))
211 simprlr 780 . . . . . . . . . . . . . . . 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)
212210, 211sstrd 4006 . . . . . . . . . . . . . . 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)
213198, 212eqsstrid 4044 . . . . . . . . . . . . . 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)
214195, 197, 213elrabd 3697 . . . . . . . . . . . . 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})
215193, 214eqeltrd 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})
216183, 215sylan2b 594 . . . . . . . . . . 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})
217216ralrimivva 3200 . . . . . . . . . 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})
218188mpofun 7557 . . . . . . . . . . 11 Fun 𝐹
219 ssrab2 4090 . . . . . . . . . . . . 13 {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇)
220 ssrab2 4090 . . . . . . . . . . . . 13 {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))} ⊆ (𝑅 Cn 𝑆)
221 xpss12 5704 . . . . . . . . . . . . 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)))
222219, 220, 221mp2an 692 . . . . . . . . . . . 12 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
223 vex 3482 . . . . . . . . . . . . . 14 𝑓 ∈ V
224 vex 3482 . . . . . . . . . . . . . 14 𝑔 ∈ V
225223, 224coex 7953 . . . . . . . . . . . . 13 (𝑓𝑔) ∈ V
226188, 225dmmpo 8095 . . . . . . . . . . . 12 dom 𝐹 = ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
227222, 226sseqtrri 4033 . . . . . . . . . . 11 ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ⊆ dom 𝐹
228 funimassov 7610 . . . . . . . . . . 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}))
229218, 227, 228mp2an 692 . . . . . . . . . 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})
230217, 229sylibr 234 . . . . . . . . 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})
231 funimass3 7074 . . . . . . . . . 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})))
232218, 227, 231mp2an 692 . . . . . . . . 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}))
233230, 232sylib 218 . . . . . . . 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}))
234 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))})))
235 sseq1 4021 . . . . . . . . . 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})))
236234, 235anbi12d 632 . . . . . . . . 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}))))
237236rspcev 3622 . . . . . . . 8 ((({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏𝐾) ⊆ 𝑦𝑤 ((int‘𝑆)‘(𝑘𝑦))}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
238135, 176, 233, 237syl12anc 837 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ ((𝐵𝐾) = 𝑤 ∧ (𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)))) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
239238expr 456 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝐵𝐾) = 𝑤) → ((𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
240239exlimdv 1931 . . . . 5 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝐵𝐾) = 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
24187, 240syldan 591 . . . 4 (((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) ∧ (𝑆t (𝐵𝐾)) = 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
242241expimpd 453 . . 3 ((𝜑𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)) → (( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
243242rexlimdva 3153 . 2 (𝜑 → (∃𝑤 ∈ (𝒫 (𝑆t (𝐵𝐾)) ∩ Fin)( (𝑆t (𝐵𝐾)) = 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (𝐴𝑉) ∧ ∀𝑦𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘𝑦)) ∧ (𝑆t (𝑘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉}))))
24484, 243mpd 15 1 (𝜑 → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605  cop 4637   cuni 4912   ciun 4996   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984  t crest 17467  Topctop 22915  intcnt 23041   Cn ccn 23248  Compccmp 23410  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cmp 23411  df-nlly 23491  df-tx 23586  df-xko 23587
This theorem is referenced by:  xkococn  23684
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