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Theorem fiuncmp 21117
Description: A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
fiuncmp.1 𝑋 = 𝐽
Assertion
Ref Expression
fiuncmp ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽
Allowed substitution hints:   𝐵(𝑥)   𝑋(𝑥)

Proof of Theorem fiuncmp
Dummy variables 𝑡 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3603 . 2 𝐴𝐴
2 simp2 1060 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → 𝐴 ∈ Fin)
3 sseq1 3605 . . . . . 6 (𝑡 = ∅ → (𝑡𝐴 ↔ ∅ ⊆ 𝐴))
4 iuneq1 4500 . . . . . . . . 9 (𝑡 = ∅ → 𝑥𝑡 𝐵 = 𝑥 ∈ ∅ 𝐵)
5 0iun 4543 . . . . . . . . 9 𝑥 ∈ ∅ 𝐵 = ∅
64, 5syl6eq 2671 . . . . . . . 8 (𝑡 = ∅ → 𝑥𝑡 𝐵 = ∅)
76oveq2d 6620 . . . . . . 7 (𝑡 = ∅ → (𝐽t 𝑥𝑡 𝐵) = (𝐽t ∅))
87eleq1d 2683 . . . . . 6 (𝑡 = ∅ → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t ∅) ∈ Comp))
93, 8imbi12d 334 . . . . 5 (𝑡 = ∅ → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (∅ ⊆ 𝐴 → (𝐽t ∅) ∈ Comp)))
109imbi2d 330 . . . 4 (𝑡 = ∅ → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (∅ ⊆ 𝐴 → (𝐽t ∅) ∈ Comp))))
11 sseq1 3605 . . . . . 6 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
12 iuneq1 4500 . . . . . . . 8 (𝑡 = 𝑦 𝑥𝑡 𝐵 = 𝑥𝑦 𝐵)
1312oveq2d 6620 . . . . . . 7 (𝑡 = 𝑦 → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥𝑦 𝐵))
1413eleq1d 2683 . . . . . 6 (𝑡 = 𝑦 → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥𝑦 𝐵) ∈ Comp))
1511, 14imbi12d 334 . . . . 5 (𝑡 = 𝑦 → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)))
1615imbi2d 330 . . . 4 (𝑡 = 𝑦 → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp))))
17 sseq1 3605 . . . . . 6 (𝑡 = (𝑦 ∪ {𝑧}) → (𝑡𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
18 iuneq1 4500 . . . . . . . 8 (𝑡 = (𝑦 ∪ {𝑧}) → 𝑥𝑡 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1918oveq2d 6620 . . . . . . 7 (𝑡 = (𝑦 ∪ {𝑧}) → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵))
2019eleq1d 2683 . . . . . 6 (𝑡 = (𝑦 ∪ {𝑧}) → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))
2117, 20imbi12d 334 . . . . 5 (𝑡 = (𝑦 ∪ {𝑧}) → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
2221imbi2d 330 . . . 4 (𝑡 = (𝑦 ∪ {𝑧}) → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))))
23 sseq1 3605 . . . . . 6 (𝑡 = 𝐴 → (𝑡𝐴𝐴𝐴))
24 iuneq1 4500 . . . . . . . 8 (𝑡 = 𝐴 𝑥𝑡 𝐵 = 𝑥𝐴 𝐵)
2524oveq2d 6620 . . . . . . 7 (𝑡 = 𝐴 → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥𝐴 𝐵))
2625eleq1d 2683 . . . . . 6 (𝑡 = 𝐴 → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥𝐴 𝐵) ∈ Comp))
2723, 26imbi12d 334 . . . . 5 (𝑡 = 𝐴 → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)))
2827imbi2d 330 . . . 4 (𝑡 = 𝐴 → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp))))
29 rest0 20883 . . . . . . 7 (𝐽 ∈ Top → (𝐽t ∅) = {∅})
30 0cmp 21107 . . . . . . 7 {∅} ∈ Comp
3129, 30syl6eqel 2706 . . . . . 6 (𝐽 ∈ Top → (𝐽t ∅) ∈ Comp)
32313ad2ant1 1080 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐽t ∅) ∈ Comp)
3332a1d 25 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (∅ ⊆ 𝐴 → (𝐽t ∅) ∈ Comp))
34 ssun1 3754 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 id 22 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
3634, 35syl5ss 3594 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑦𝐴)
3736imim1i 63 . . . . . . 7 ((𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp))
38 simpl1 1062 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝐽 ∈ Top)
39 iunxun 4571 . . . . . . . . . . . 12 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
40 simprr 795 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)
41 cmptop 21108 . . . . . . . . . . . . . 14 ((𝐽t 𝑥𝑦 𝐵) ∈ Comp → (𝐽t 𝑥𝑦 𝐵) ∈ Top)
42 restrcl 20871 . . . . . . . . . . . . . . 15 ((𝐽t 𝑥𝑦 𝐵) ∈ Top → (𝐽 ∈ V ∧ 𝑥𝑦 𝐵 ∈ V))
4342simprd 479 . . . . . . . . . . . . . 14 ((𝐽t 𝑥𝑦 𝐵) ∈ Top → 𝑥𝑦 𝐵 ∈ V)
4440, 41, 433syl 18 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑥𝑦 𝐵 ∈ V)
45 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑡𝐵
46 nfcsb1v 3530 . . . . . . . . . . . . . . . 16 𝑥𝑡 / 𝑥𝐵
47 csbeq1a 3523 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡𝐵 = 𝑡 / 𝑥𝐵)
4845, 46, 47cbviun 4523 . . . . . . . . . . . . . . 15 𝑥 ∈ {𝑧}𝐵 = 𝑡 ∈ {𝑧}𝑡 / 𝑥𝐵
49 vex 3189 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
50 csbeq1 3517 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧𝑡 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5149, 50iunxsn 4569 . . . . . . . . . . . . . . 15 𝑡 ∈ {𝑧}𝑡 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
5248, 51eqtri 2643 . . . . . . . . . . . . . 14 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
53 ssun2 3755 . . . . . . . . . . . . . . . . . 18 {𝑧} ⊆ (𝑦 ∪ {𝑧})
54 simprl 793 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
5553, 54syl5ss 3594 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → {𝑧} ⊆ 𝐴)
5649snss 4286 . . . . . . . . . . . . . . . . 17 (𝑧𝐴 ↔ {𝑧} ⊆ 𝐴)
5755, 56sylibr 224 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑧𝐴)
58 simpl3 1064 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp)
59 nfv 1840 . . . . . . . . . . . . . . . . . 18 𝑡(𝐽t 𝐵) ∈ Comp
60 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐽
61 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑥t
6260, 61, 46nfov 6630 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐽t 𝑡 / 𝑥𝐵)
6362nfel1 2775 . . . . . . . . . . . . . . . . . 18 𝑥(𝐽t 𝑡 / 𝑥𝐵) ∈ Comp
6447oveq2d 6620 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑡 → (𝐽t 𝐵) = (𝐽t 𝑡 / 𝑥𝐵))
6564eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝐽t 𝐵) ∈ Comp ↔ (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp))
6659, 63, 65cbvral 3155 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp ↔ ∀𝑡𝐴 (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp)
6758, 66sylib 208 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ∀𝑡𝐴 (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp)
6850oveq2d 6620 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑧 → (𝐽t 𝑡 / 𝑥𝐵) = (𝐽t 𝑧 / 𝑥𝐵))
6968eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑧 → ((𝐽t 𝑡 / 𝑥𝐵) ∈ Comp ↔ (𝐽t 𝑧 / 𝑥𝐵) ∈ Comp))
7069rspcv 3291 . . . . . . . . . . . . . . . 16 (𝑧𝐴 → (∀𝑡𝐴 (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp → (𝐽t 𝑧 / 𝑥𝐵) ∈ Comp))
7157, 67, 70sylc 65 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑧 / 𝑥𝐵) ∈ Comp)
72 cmptop 21108 . . . . . . . . . . . . . . 15 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Comp → (𝐽t 𝑧 / 𝑥𝐵) ∈ Top)
73 restrcl 20871 . . . . . . . . . . . . . . . 16 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Top → (𝐽 ∈ V ∧ 𝑧 / 𝑥𝐵 ∈ V))
7473simprd 479 . . . . . . . . . . . . . . 15 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Top → 𝑧 / 𝑥𝐵 ∈ V)
7571, 72, 743syl 18 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑧 / 𝑥𝐵 ∈ V)
7652, 75syl5eqel 2702 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑥 ∈ {𝑧}𝐵 ∈ V)
77 unexg 6912 . . . . . . . . . . . . 13 (( 𝑥𝑦 𝐵 ∈ V ∧ 𝑥 ∈ {𝑧}𝐵 ∈ V) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ V)
7844, 76, 77syl2anc 692 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ V)
7939, 78syl5eqel 2702 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V)
80 resttop 20874 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
8138, 79, 80syl2anc 692 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
82 eqid 2621 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8382restin 20880 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
8438, 79, 83syl2anc 692 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
8584unieqd 4412 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
86 inss2 3812 . . . . . . . . . . . . . 14 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝐽
87 fiuncmp.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
8886, 87sseqtr4i 3617 . . . . . . . . . . . . 13 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝑋
8987restuni 20876 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝑋) → ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
9038, 88, 89sylancl 693 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
9185, 90eqtr4d 2658 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽))
9252uneq2i 3742 . . . . . . . . . . . . . 14 ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) = ( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵)
9339, 92eqtri 2643 . . . . . . . . . . . . 13 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵)
9493ineq1i 3788 . . . . . . . . . . . 12 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵) ∩ 𝐽)
95 indir 3851 . . . . . . . . . . . 12 (( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵) ∩ 𝐽) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽))
9694, 95eqtri 2643 . . . . . . . . . . 11 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽))
9791, 96syl6eq 2671 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽)))
98 inss1 3811 . . . . . . . . . . . . . . 15 ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥𝑦 𝐵
99 ssun1 3754 . . . . . . . . . . . . . . . 16 𝑥𝑦 𝐵 ⊆ ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
10099, 39sseqtr4i 3617 . . . . . . . . . . . . . . 15 𝑥𝑦 𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
10198, 100sstri 3592 . . . . . . . . . . . . . 14 ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
102101a1i 11 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
103 restabs 20879 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t ( 𝑥𝑦 𝐵 𝐽)) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
10438, 102, 79, 103syl3anc 1323 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t ( 𝑥𝑦 𝐵 𝐽)) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
10582restin 20880 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑥𝑦 𝐵 ∈ V) → (𝐽t 𝑥𝑦 𝐵) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
10638, 44, 105syl2anc 692 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥𝑦 𝐵) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
107104, 106eqtr4d 2658 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t ( 𝑥𝑦 𝐵 𝐽)) = (𝐽t 𝑥𝑦 𝐵))
108107, 40eqeltrd 2698 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t ( 𝑥𝑦 𝐵 𝐽)) ∈ Comp)
109 inss1 3811 . . . . . . . . . . . . . . 15 (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑧 / 𝑥𝐵
110 ssun2 3755 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {𝑧}𝐵 ⊆ ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
111110, 39sseqtr4i 3617 . . . . . . . . . . . . . . . 16 𝑥 ∈ {𝑧}𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
11252, 111eqsstr3i 3615 . . . . . . . . . . . . . . 15 𝑧 / 𝑥𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
113109, 112sstri 3592 . . . . . . . . . . . . . 14 (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
114113a1i 11 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
115 restabs 20879 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t (𝑧 / 𝑥𝐵 𝐽)) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
11638, 114, 79, 115syl3anc 1323 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t (𝑧 / 𝑥𝐵 𝐽)) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
11782restin 20880 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑧 / 𝑥𝐵 ∈ V) → (𝐽t 𝑧 / 𝑥𝐵) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
11838, 75, 117syl2anc 692 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑧 / 𝑥𝐵) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
119116, 118eqtr4d 2658 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t (𝑧 / 𝑥𝐵 𝐽)) = (𝐽t 𝑧 / 𝑥𝐵))
120119, 71eqeltrd 2698 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t (𝑧 / 𝑥𝐵 𝐽)) ∈ Comp)
121 eqid 2621 . . . . . . . . . . 11 (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
122121uncmp 21116 . . . . . . . . . 10 ((((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top ∧ (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽))) ∧ (((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t ( 𝑥𝑦 𝐵 𝐽)) ∈ Comp ∧ ((𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾t (𝑧 / 𝑥𝐵 𝐽)) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)
12381, 97, 108, 120, 122syl22anc 1324 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)
124123exp32 630 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐽t 𝑥𝑦 𝐵) ∈ Comp → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
125124a2d 29 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
12637, 125syl5 34 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
127126a2i 14 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
128127a1i 11 . . . 4 (𝑦 ∈ Fin → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))))
12910, 16, 22, 28, 33, 128findcard2 8144 . . 3 (𝐴 ∈ Fin → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)))
1302, 129mpcom 38 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp))
1311, 130mpi 20 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  csb 3514  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148   cuni 4402   ciun 4485  (class class class)co 6604  Fincfn 7899  t crest 16002  Topctop 20617  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100
This theorem is referenced by:  xkococnlem  21372
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