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Theorem cmpsub 21184
Description: Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Hypothesis
Ref Expression
cmpsub.1 𝑋 = 𝐽
Assertion
Ref Expression
cmpsub ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
Distinct variable groups:   𝑐,𝑑,𝐽   𝑆,𝑐,𝑑   𝑋,𝑐,𝑑

Proof of Theorem cmpsub
Dummy variables 𝑥 𝑦 𝑓 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2620 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
21iscmp 21172 . . 3 ((𝐽t 𝑆) ∈ Comp ↔ ((𝐽t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))
3 id 22 . . . . . 6 (𝑆𝑋𝑆𝑋)
4 cmpsub.1 . . . . . . 7 𝑋 = 𝐽
54topopn 20692 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
6 ssexg 4795 . . . . . 6 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
73, 5, 6syl2anr 495 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
8 resttop 20945 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
97, 8syldan 487 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
10 ibar 525 . . . . 5 ((𝐽t 𝑆) ∈ Top → (∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) ↔ ((𝐽t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡))))
1110bicomd 213 . . . 4 ((𝐽t 𝑆) ∈ Top → (((𝐽t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)) ↔ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))
129, 11syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((𝐽t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)) ↔ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))
132, 12syl5bb 272 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))
14 vex 3198 . . . . . . . . . . 11 𝑡 ∈ V
15 eqeq1 2624 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑥 = (𝑦𝑆) ↔ 𝑡 = (𝑦𝑆)))
1615rexbidv 3048 . . . . . . . . . . 11 (𝑥 = 𝑡 → (∃𝑦𝑐 𝑥 = (𝑦𝑆) ↔ ∃𝑦𝑐 𝑡 = (𝑦𝑆)))
1714, 16elab 3344 . . . . . . . . . 10 (𝑡 ∈ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ ∃𝑦𝑐 𝑡 = (𝑦𝑆))
18 selpw 4156 . . . . . . . . . . . . . 14 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
19 ssel2 3590 . . . . . . . . . . . . . . . 16 ((𝑐𝐽𝑦𝑐) → 𝑦𝐽)
20 ineq1 3799 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑦 → (𝑑𝑆) = (𝑦𝑆))
2120eqeq2d 2630 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑦 → (𝑡 = (𝑑𝑆) ↔ 𝑡 = (𝑦𝑆)))
2221rspcev 3304 . . . . . . . . . . . . . . . . 17 ((𝑦𝐽𝑡 = (𝑦𝑆)) → ∃𝑑𝐽 𝑡 = (𝑑𝑆))
2322ex 450 . . . . . . . . . . . . . . . 16 (𝑦𝐽 → (𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆)))
2419, 23syl 17 . . . . . . . . . . . . . . 15 ((𝑐𝐽𝑦𝑐) → (𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆)))
2524ex 450 . . . . . . . . . . . . . 14 (𝑐𝐽 → (𝑦𝑐 → (𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆))))
2618, 25sylbi 207 . . . . . . . . . . . . 13 (𝑐 ∈ 𝒫 𝐽 → (𝑦𝑐 → (𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆))))
2726adantl 482 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑦𝑐 → (𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆))))
2827rexlimdv 3026 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∃𝑦𝑐 𝑡 = (𝑦𝑆) → ∃𝑑𝐽 𝑡 = (𝑑𝑆)))
29 simpll 789 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → 𝐽 ∈ Top)
304sseq2i 3622 . . . . . . . . . . . . . 14 (𝑆𝑋𝑆 𝐽)
31 uniexg 6940 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → 𝐽 ∈ V)
32 ssexg 4795 . . . . . . . . . . . . . . . 16 ((𝑆 𝐽 𝐽 ∈ V) → 𝑆 ∈ V)
3331, 32sylan2 491 . . . . . . . . . . . . . . 15 ((𝑆 𝐽𝐽 ∈ Top) → 𝑆 ∈ V)
3433ancoms 469 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ∈ V)
3530, 34sylan2b 492 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
3635adantr 481 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → 𝑆 ∈ V)
37 elrest 16069 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝑡 ∈ (𝐽t 𝑆) ↔ ∃𝑑𝐽 𝑡 = (𝑑𝑆)))
3829, 36, 37syl2anc 692 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑡 ∈ (𝐽t 𝑆) ↔ ∃𝑑𝐽 𝑡 = (𝑑𝑆)))
3928, 38sylibrd 249 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∃𝑦𝑐 𝑡 = (𝑦𝑆) → 𝑡 ∈ (𝐽t 𝑆)))
4017, 39syl5bi 232 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑡 ∈ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → 𝑡 ∈ (𝐽t 𝑆)))
4140ssrdv 3601 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ⊆ (𝐽t 𝑆))
42 vex 3198 . . . . . . . . . 10 𝑐 ∈ V
4342abrexex 7126 . . . . . . . . 9 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∈ V
4443elpw 4155 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∈ 𝒫 (𝐽t 𝑆) ↔ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ⊆ (𝐽t 𝑆))
4541, 44sylibr 224 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∈ 𝒫 (𝐽t 𝑆))
46 unieq 4435 . . . . . . . . . 10 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → 𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)})
4746eqeq2d 2630 . . . . . . . . 9 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ( (𝐽t 𝑆) = 𝑠 (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)}))
48 pweq 4152 . . . . . . . . . . 11 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → 𝒫 𝑠 = 𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)})
4948ineq1d 3805 . . . . . . . . . 10 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → (𝒫 𝑠 ∩ Fin) = (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin))
5049rexeqdv 3140 . . . . . . . . 9 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → (∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡 ↔ ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡))
5147, 50imbi12d 334 . . . . . . . 8 (𝑠 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → (( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) ↔ ( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡)))
5251rspcva 3302 . . . . . . 7 (({𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∈ 𝒫 (𝐽t 𝑆) ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)) → ( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡))
5345, 52sylan 488 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)) → ( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡))
5453ex 450 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) → ( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡)))
554restuni 20947 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
5655ad2antrr 761 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → 𝑆 = (𝐽t 𝑆))
57 vex 3198 . . . . . . . . . . . . . 14 𝑦 ∈ V
5857inex1 4790 . . . . . . . . . . . . 13 (𝑦𝑆) ∈ V
5958dfiun2 4545 . . . . . . . . . . . 12 𝑦𝑐 (𝑦𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)}
60 incom 3797 . . . . . . . . . . . . . 14 (𝑦𝑆) = (𝑆𝑦)
6160a1i 11 . . . . . . . . . . . . 13 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ 𝑦𝑐) → (𝑦𝑆) = (𝑆𝑦))
6261iuneq2dv 4533 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → 𝑦𝑐 (𝑦𝑆) = 𝑦𝑐 (𝑆𝑦))
6359, 62syl5eqr 2668 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} = 𝑦𝑐 (𝑆𝑦))
64 iunin2 4575 . . . . . . . . . . . 12 𝑦𝑐 (𝑆𝑦) = (𝑆 𝑦𝑐 𝑦)
65 uniiun 4564 . . . . . . . . . . . . . . . 16 𝑐 = 𝑦𝑐 𝑦
6665eqcomi 2629 . . . . . . . . . . . . . . 15 𝑦𝑐 𝑦 = 𝑐
6766a1i 11 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → 𝑦𝑐 𝑦 = 𝑐)
6867ineq2d 3806 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (𝑆 𝑦𝑐 𝑦) = (𝑆 𝑐))
69 incom 3797 . . . . . . . . . . . . . . 15 (𝑆 𝑐) = ( 𝑐𝑆)
70 sseqin2 3809 . . . . . . . . . . . . . . . 16 (𝑆 𝑐 ↔ ( 𝑐𝑆) = 𝑆)
7170biimpi 206 . . . . . . . . . . . . . . 15 (𝑆 𝑐 → ( 𝑐𝑆) = 𝑆)
7269, 71syl5eq 2666 . . . . . . . . . . . . . 14 (𝑆 𝑐 → (𝑆 𝑐) = 𝑆)
7372adantl 482 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (𝑆 𝑐) = 𝑆)
7468, 73eqtrd 2654 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (𝑆 𝑦𝑐 𝑦) = 𝑆)
7564, 74syl5eq 2666 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → 𝑦𝑐 (𝑆𝑦) = 𝑆)
7663, 75eqtr2d 2655 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → 𝑆 = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)})
7756, 76eqeq12d 2635 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (𝑆 = 𝑆 (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)}))
7856eqeq1d 2622 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (𝑆 = 𝑡 (𝐽t 𝑆) = 𝑡))
7978rexbidv 3048 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡 ↔ ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡))
8077, 79imbi12d 334 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → ((𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡) ↔ ( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡)))
81 eqid 2620 . . . . . . . . . 10 𝑆 = 𝑆
8281a1bi 352 . . . . . . . . 9 (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡 ↔ (𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡))
83 elin 3788 . . . . . . . . . . . 12 (𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) ↔ (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∧ 𝑡 ∈ Fin))
84 selpw 4156 . . . . . . . . . . . . . 14 (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ 𝑡 ⊆ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)})
85 dfss3 3585 . . . . . . . . . . . . . 14 (𝑡 ⊆ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ ∀𝑠𝑡 𝑠 ∈ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)})
86 vex 3198 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
87 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑠 → (𝑥 = (𝑦𝑆) ↔ 𝑠 = (𝑦𝑆)))
8887rexbidv 3048 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑠 → (∃𝑦𝑐 𝑥 = (𝑦𝑆) ↔ ∃𝑦𝑐 𝑠 = (𝑦𝑆)))
8986, 88elab 3344 . . . . . . . . . . . . . . 15 (𝑠 ∈ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ ∃𝑦𝑐 𝑠 = (𝑦𝑆))
9089ralbii 2977 . . . . . . . . . . . . . 14 (∀𝑠𝑡 𝑠 ∈ {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ ∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆))
9184, 85, 903bitri 286 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ↔ ∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆))
9291anbi1i 730 . . . . . . . . . . . 12 ((𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∧ 𝑡 ∈ Fin) ↔ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin))
9383, 92bitri 264 . . . . . . . . . . 11 (𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) ↔ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin))
94 ineq1 3799 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑓𝑠) → (𝑦𝑆) = ((𝑓𝑠) ∩ 𝑆))
9594eqeq2d 2630 . . . . . . . . . . . . . . 15 (𝑦 = (𝑓𝑠) → (𝑠 = (𝑦𝑆) ↔ 𝑠 = ((𝑓𝑠) ∩ 𝑆)))
9695ac6sfi 8189 . . . . . . . . . . . . . 14 ((𝑡 ∈ Fin ∧ ∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆)) → ∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)))
9796ancoms 469 . . . . . . . . . . . . 13 ((∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin) → ∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)))
9897adantl 482 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → ∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)))
99 frn 6040 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝑡𝑐 → ran 𝑓𝑐)
10099ad2antrl 763 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → ran 𝑓𝑐)
101 vex 3198 . . . . . . . . . . . . . . . . . . . . . 22 𝑓 ∈ V
102101rnex 7085 . . . . . . . . . . . . . . . . . . . . 21 ran 𝑓 ∈ V
103102elpw 4155 . . . . . . . . . . . . . . . . . . . 20 (ran 𝑓 ∈ 𝒫 𝑐 ↔ ran 𝑓𝑐)
104100, 103sylibr 224 . . . . . . . . . . . . . . . . . . 19 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → ran 𝑓 ∈ 𝒫 𝑐)
105 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → 𝑡 ∈ Fin)
106105ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → 𝑡 ∈ Fin)
107 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓:𝑡𝑐𝑓 Fn 𝑡)
108 dffn4 6108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
109107, 108sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:𝑡𝑐𝑓:𝑡onto→ran 𝑓)
110 fodomfi 8224 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓𝑡)
111109, 110sylan2 491 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ Fin ∧ 𝑓:𝑡𝑐) → ran 𝑓𝑡)
112111adantll 749 . . . . . . . . . . . . . . . . . . . . . 22 (((∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin) ∧ 𝑓:𝑡𝑐) → ran 𝑓𝑡)
113112adantll 749 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑓:𝑡𝑐) → ran 𝑓𝑡)
114113ad2ant2r 782 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → ran 𝑓𝑡)
115 domfi 8166 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ Fin ∧ ran 𝑓𝑡) → ran 𝑓 ∈ Fin)
116106, 114, 115syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → ran 𝑓 ∈ Fin)
117104, 116elind 3790 . . . . . . . . . . . . . . . . . 18 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
118 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = 𝑢𝑠 = 𝑢)
119 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 = 𝑢 → (𝑓𝑠) = (𝑓𝑢))
120119ineq1d 3805 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = 𝑢 → ((𝑓𝑠) ∩ 𝑆) = ((𝑓𝑢) ∩ 𝑆))
121118, 120eqeq12d 2635 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 = 𝑢 → (𝑠 = ((𝑓𝑠) ∩ 𝑆) ↔ 𝑢 = ((𝑓𝑢) ∩ 𝑆)))
122121rspccv 3301 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆) → (𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)))
123 pm2.27 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑢𝑡 → ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)) → 𝑢 = ((𝑓𝑢) ∩ 𝑆)))
124 inss1 3825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓𝑢) ∩ 𝑆) ⊆ (𝑓𝑢)
125 sseq1 3618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑢 = ((𝑓𝑢) ∩ 𝑆) → (𝑢 ⊆ (𝑓𝑢) ↔ ((𝑓𝑢) ∩ 𝑆) ⊆ (𝑓𝑢)))
126124, 125mpbiri 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑢 = ((𝑓𝑢) ∩ 𝑆) → 𝑢 ⊆ (𝑓𝑢))
127 ssel 3589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑢 ⊆ (𝑓𝑢) → (𝑤𝑢𝑤 ∈ (𝑓𝑢)))
128127a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑢 ⊆ (𝑓𝑢) → (𝑤𝑢 → (𝑓:𝑡𝑐𝑤 ∈ (𝑓𝑢))))
129126, 128syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑢 = ((𝑓𝑢) ∩ 𝑆) → (𝑤𝑢 → (𝑓:𝑡𝑐𝑤 ∈ (𝑓𝑢))))
130129a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑢𝑡 → (𝑢 = ((𝑓𝑢) ∩ 𝑆) → (𝑤𝑢 → (𝑓:𝑡𝑐𝑤 ∈ (𝑓𝑢)))))
1311303imp 1254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆) ∧ 𝑤𝑢) → (𝑓:𝑡𝑐𝑤 ∈ (𝑓𝑢)))
132 fnfvelrn 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑓 Fn 𝑡𝑢𝑡) → (𝑓𝑢) ∈ ran 𝑓)
133132expcom 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑢𝑡 → (𝑓 Fn 𝑡 → (𝑓𝑢) ∈ ran 𝑓))
1341333ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆) ∧ 𝑤𝑢) → (𝑓 Fn 𝑡 → (𝑓𝑢) ∈ ran 𝑓))
135107, 134syl5 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆) ∧ 𝑤𝑢) → (𝑓:𝑡𝑐 → (𝑓𝑢) ∈ ran 𝑓))
136131, 135jcad 555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆) ∧ 𝑤𝑢) → (𝑓:𝑡𝑐 → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))
1371363exp 1262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑢𝑡 → (𝑢 = ((𝑓𝑢) ∩ 𝑆) → (𝑤𝑢 → (𝑓:𝑡𝑐 → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))))
138123, 137syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢𝑡 → ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)) → (𝑤𝑢 → (𝑓:𝑡𝑐 → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))))
139138com3r 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤𝑢 → (𝑢𝑡 → ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)) → (𝑓:𝑡𝑐 → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))))
140139imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑤𝑢𝑢𝑡) → ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)) → (𝑓:𝑡𝑐 → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓))))
141140com3l 89 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆)) → (𝑓:𝑡𝑐 → ((𝑤𝑢𝑢𝑡) → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓))))
142141impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑡𝑐 ∧ (𝑢𝑡𝑢 = ((𝑓𝑢) ∩ 𝑆))) → ((𝑤𝑢𝑢𝑡) → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))
143122, 142sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → ((𝑤𝑢𝑢𝑡) → (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))
144 fvex 6188 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓𝑢) ∈ V
145 eleq2 2688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = (𝑓𝑢) → (𝑤𝑣𝑤 ∈ (𝑓𝑢)))
146 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = (𝑓𝑢) → (𝑣 ∈ ran 𝑓 ↔ (𝑓𝑢) ∈ ran 𝑓))
147145, 146anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = (𝑓𝑢) → ((𝑤𝑣𝑣 ∈ ran 𝑓) ↔ (𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓)))
148144, 147spcev 3295 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 ∈ (𝑓𝑢) ∧ (𝑓𝑢) ∈ ran 𝑓) → ∃𝑣(𝑤𝑣𝑣 ∈ ran 𝑓))
149143, 148syl6 35 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → ((𝑤𝑢𝑢𝑡) → ∃𝑣(𝑤𝑣𝑣 ∈ ran 𝑓)))
150149exlimdv 1859 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → (∃𝑢(𝑤𝑢𝑢𝑡) → ∃𝑣(𝑤𝑣𝑣 ∈ ran 𝑓)))
151 eluni 4430 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 𝑡 ↔ ∃𝑢(𝑤𝑢𝑢𝑡))
152 eluni 4430 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ran 𝑓 ↔ ∃𝑣(𝑤𝑣𝑣 ∈ ran 𝑓))
153150, 151, 1523imtr4g 285 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → (𝑤 𝑡𝑤 ran 𝑓))
154153ssrdv 3601 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → 𝑡 ran 𝑓)
155154adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → 𝑡 ran 𝑓)
156 sseq1 3618 . . . . . . . . . . . . . . . . . . . 20 (𝑆 = 𝑡 → (𝑆 ran 𝑓 𝑡 ran 𝑓))
157156ad2antlr 762 . . . . . . . . . . . . . . . . . . 19 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → (𝑆 ran 𝑓 𝑡 ran 𝑓))
158155, 157mpbird 247 . . . . . . . . . . . . . . . . . 18 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → 𝑆 ran 𝑓)
159117, 158jca 554 . . . . . . . . . . . . . . . . 17 (((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) ∧ (𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆))) → (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓))
160159ex 450 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) → ((𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓)))
161160eximdv 1844 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = 𝑡) → (∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓)))
162161ex 450 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → (𝑆 = 𝑡 → (∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓))))
163162com23 86 . . . . . . . . . . . . 13 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → (∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → (𝑆 = 𝑡 → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓))))
164 unieq 4435 . . . . . . . . . . . . . . . 16 (𝑑 = ran 𝑓 𝑑 = ran 𝑓)
165164sseq2d 3625 . . . . . . . . . . . . . . 15 (𝑑 = ran 𝑓 → (𝑆 𝑑𝑆 ran 𝑓))
166165rspcev 3304 . . . . . . . . . . . . . 14 ((ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)
167166exlimiv 1856 . . . . . . . . . . . . 13 (∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)
168163, 167syl8 76 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → (∃𝑓(𝑓:𝑡𝑐 ∧ ∀𝑠𝑡 𝑠 = ((𝑓𝑠) ∩ 𝑆)) → (𝑆 = 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
16998, 168mpd 15 . . . . . . . . . . 11 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ (∀𝑠𝑡𝑦𝑐 𝑠 = (𝑦𝑆) ∧ 𝑡 ∈ Fin)) → (𝑆 = 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑))
17093, 169sylan2b 492 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) ∧ 𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)) → (𝑆 = 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑))
171170rexlimdva 3027 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑))
17282, 171syl5bir 233 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → ((𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin)𝑆 = 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑))
17380, 172sylbird 250 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 𝑐) → (( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑))
174173ex 450 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑆 𝑐 → (( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
175174com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (( (𝐽t 𝑆) = {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦𝑐 𝑥 = (𝑦𝑆)} ∩ Fin) (𝐽t 𝑆) = 𝑡) → (𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
17654, 175syld 47 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) → (𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
177176ralrimdva 2966 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) → ∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
1784cmpsublem 21183 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))
179177, 178impbid 202 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡) ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
18013, 179bitrd 268 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  𝒫 cpw 4149   cuni 4427   ciun 4511   class class class wbr 4644  ran crn 5105   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876  (class class class)co 6635  cdom 7938  Fincfn 7940  t crest 16062  Topctop 20679  Compccmp 21170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-fin 7944  df-fi 8302  df-rest 16064  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cmp 21171
This theorem is referenced by:  cmpcld  21186  uncmp  21187  hauscmplem  21190  1stckgenlem  21337  icccmp  22609  bndth  22738  ovolicc2  23271  stoweidlem50  40030  stoweidlem57  40037
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