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Theorem fiuncmp 21409
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 3765 . 2 𝐴𝐴
2 simp2 1132 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → 𝐴 ∈ Fin)
3 sseq1 3767 . . . . . 6 (𝑡 = ∅ → (𝑡𝐴 ↔ ∅ ⊆ 𝐴))
4 iuneq1 4686 . . . . . . . . 9 (𝑡 = ∅ → 𝑥𝑡 𝐵 = 𝑥 ∈ ∅ 𝐵)
5 0iun 4729 . . . . . . . . 9 𝑥 ∈ ∅ 𝐵 = ∅
64, 5syl6eq 2810 . . . . . . . 8 (𝑡 = ∅ → 𝑥𝑡 𝐵 = ∅)
76oveq2d 6829 . . . . . . 7 (𝑡 = ∅ → (𝐽t 𝑥𝑡 𝐵) = (𝐽t ∅))
87eleq1d 2824 . . . . . 6 (𝑡 = ∅ → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t ∅) ∈ Comp))
93, 8imbi12d 333 . . . . 5 (𝑡 = ∅ → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (∅ ⊆ 𝐴 → (𝐽t ∅) ∈ Comp)))
109imbi2d 329 . . . 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 3767 . . . . . 6 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
12 iuneq1 4686 . . . . . . . 8 (𝑡 = 𝑦 𝑥𝑡 𝐵 = 𝑥𝑦 𝐵)
1312oveq2d 6829 . . . . . . 7 (𝑡 = 𝑦 → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥𝑦 𝐵))
1413eleq1d 2824 . . . . . 6 (𝑡 = 𝑦 → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥𝑦 𝐵) ∈ Comp))
1511, 14imbi12d 333 . . . . 5 (𝑡 = 𝑦 → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)))
1615imbi2d 329 . . . 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 3767 . . . . . 6 (𝑡 = (𝑦 ∪ {𝑧}) → (𝑡𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
18 iuneq1 4686 . . . . . . . 8 (𝑡 = (𝑦 ∪ {𝑧}) → 𝑥𝑡 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1918oveq2d 6829 . . . . . . 7 (𝑡 = (𝑦 ∪ {𝑧}) → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵))
2019eleq1d 2824 . . . . . 6 (𝑡 = (𝑦 ∪ {𝑧}) → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))
2117, 20imbi12d 333 . . . . 5 (𝑡 = (𝑦 ∪ {𝑧}) → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
2221imbi2d 329 . . . 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 3767 . . . . . 6 (𝑡 = 𝐴 → (𝑡𝐴𝐴𝐴))
24 iuneq1 4686 . . . . . . . 8 (𝑡 = 𝐴 𝑥𝑡 𝐵 = 𝑥𝐴 𝐵)
2524oveq2d 6829 . . . . . . 7 (𝑡 = 𝐴 → (𝐽t 𝑥𝑡 𝐵) = (𝐽t 𝑥𝐴 𝐵))
2625eleq1d 2824 . . . . . 6 (𝑡 = 𝐴 → ((𝐽t 𝑥𝑡 𝐵) ∈ Comp ↔ (𝐽t 𝑥𝐴 𝐵) ∈ Comp))
2723, 26imbi12d 333 . . . . 5 (𝑡 = 𝐴 → ((𝑡𝐴 → (𝐽t 𝑥𝑡 𝐵) ∈ Comp) ↔ (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)))
2827imbi2d 329 . . . 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 21175 . . . . . . 7 (𝐽 ∈ Top → (𝐽t ∅) = {∅})
30 0cmp 21399 . . . . . . 7 {∅} ∈ Comp
3129, 30syl6eqel 2847 . . . . . 6 (𝐽 ∈ Top → (𝐽t ∅) ∈ Comp)
32313ad2ant1 1128 . . . . 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 3919 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 id 22 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
3634, 35syl5ss 3755 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑦𝐴)
3736imim1i 63 . . . . . . 7 ((𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp))
38 simpl1 1228 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝐽 ∈ Top)
39 iunxun 4757 . . . . . . . . . . . 12 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
40 simprr 813 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥𝑦 𝐵) ∈ Comp)
41 cmptop 21400 . . . . . . . . . . . . . 14 ((𝐽t 𝑥𝑦 𝐵) ∈ Comp → (𝐽t 𝑥𝑦 𝐵) ∈ Top)
42 restrcl 21163 . . . . . . . . . . . . . . 15 ((𝐽t 𝑥𝑦 𝐵) ∈ Top → (𝐽 ∈ V ∧ 𝑥𝑦 𝐵 ∈ V))
4342simprd 482 . . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . . 16 𝑡𝐵
46 nfcsb1v 3690 . . . . . . . . . . . . . . . 16 𝑥𝑡 / 𝑥𝐵
47 csbeq1a 3683 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡𝐵 = 𝑡 / 𝑥𝐵)
4845, 46, 47cbviun 4709 . . . . . . . . . . . . . . 15 𝑥 ∈ {𝑧}𝐵 = 𝑡 ∈ {𝑧}𝑡 / 𝑥𝐵
49 vex 3343 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
50 csbeq1 3677 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧𝑡 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5149, 50iunxsn 4755 . . . . . . . . . . . . . . 15 𝑡 ∈ {𝑧}𝑡 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
5248, 51eqtri 2782 . . . . . . . . . . . . . 14 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
5350oveq2d 6829 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑧 → (𝐽t 𝑡 / 𝑥𝐵) = (𝐽t 𝑧 / 𝑥𝐵))
5453eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → ((𝐽t 𝑡 / 𝑥𝐵) ∈ Comp ↔ (𝐽t 𝑧 / 𝑥𝐵) ∈ Comp))
55 simpl3 1232 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp)
56 nfv 1992 . . . . . . . . . . . . . . . . . 18 𝑡(𝐽t 𝐵) ∈ Comp
57 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐽
58 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 𝑥t
5957, 58, 46nfov 6839 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐽t 𝑡 / 𝑥𝐵)
6059nfel1 2917 . . . . . . . . . . . . . . . . . 18 𝑥(𝐽t 𝑡 / 𝑥𝐵) ∈ Comp
6147oveq2d 6829 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑡 → (𝐽t 𝐵) = (𝐽t 𝑡 / 𝑥𝐵))
6261eleq1d 2824 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝐽t 𝐵) ∈ Comp ↔ (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp))
6356, 60, 62cbvral 3306 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp ↔ ∀𝑡𝐴 (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp)
6455, 63sylib 208 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ∀𝑡𝐴 (𝐽t 𝑡 / 𝑥𝐵) ∈ Comp)
65 ssun2 3920 . . . . . . . . . . . . . . . . . 18 {𝑧} ⊆ (𝑦 ∪ {𝑧})
66 simprl 811 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
6765, 66syl5ss 3755 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → {𝑧} ⊆ 𝐴)
6849snss 4460 . . . . . . . . . . . . . . . . 17 (𝑧𝐴 ↔ {𝑧} ⊆ 𝐴)
6967, 68sylibr 224 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑧𝐴)
7054, 64, 69rspcdva 3455 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑧 / 𝑥𝐵) ∈ Comp)
71 cmptop 21400 . . . . . . . . . . . . . . 15 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Comp → (𝐽t 𝑧 / 𝑥𝐵) ∈ Top)
72 restrcl 21163 . . . . . . . . . . . . . . . 16 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Top → (𝐽 ∈ V ∧ 𝑧 / 𝑥𝐵 ∈ V))
7372simprd 482 . . . . . . . . . . . . . . 15 ((𝐽t 𝑧 / 𝑥𝐵) ∈ Top → 𝑧 / 𝑥𝐵 ∈ V)
7470, 71, 733syl 18 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑧 / 𝑥𝐵 ∈ V)
7552, 74syl5eqel 2843 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑥 ∈ {𝑧}𝐵 ∈ V)
76 unexg 7124 . . . . . . . . . . . . 13 (( 𝑥𝑦 𝐵 ∈ V ∧ 𝑥 ∈ {𝑧}𝐵 ∈ V) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ V)
7744, 75, 76syl2anc 696 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ V)
7839, 77syl5eqel 2843 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V)
79 resttop 21166 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
8038, 78, 79syl2anc 696 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
81 eqid 2760 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8281restin 21172 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
8338, 78, 82syl2anc 696 . . . . . . . . . . . . 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)))
8483unieqd 4598 . . . . . . . . . . . 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)))
85 inss2 3977 . . . . . . . . . . . . . 14 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝐽
86 fiuncmp.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
8785, 86sseqtr4i 3779 . . . . . . . . . . . . 13 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝑋
8886restuni 21168 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) ⊆ 𝑋) → ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (𝐽t ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽)))
8938, 87, 88sylancl 697 . . . . . . . . . . . 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)))
9084, 89eqtr4d 2797 . . . . . . . . . . 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))
9152uneq2i 3907 . . . . . . . . . . . . . 14 ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) = ( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵)
9239, 91eqtri 2782 . . . . . . . . . . . . 13 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵)
9392ineq1i 3953 . . . . . . . . . . . 12 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵) ∩ 𝐽)
94 indir 4018 . . . . . . . . . . . 12 (( 𝑥𝑦 𝐵𝑧 / 𝑥𝐵) ∩ 𝐽) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽))
9593, 94eqtri 2782 . . . . . . . . . . 11 ( 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 𝐽) = (( 𝑥𝑦 𝐵 𝐽) ∪ (𝑧 / 𝑥𝐵 𝐽))
9690, 95syl6eq 2810 . . . . . . . . . 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)))
97 inss1 3976 . . . . . . . . . . . . . . 15 ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥𝑦 𝐵
98 ssun1 3919 . . . . . . . . . . . . . . . 16 𝑥𝑦 𝐵 ⊆ ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
9998, 39sseqtr4i 3779 . . . . . . . . . . . . . . 15 𝑥𝑦 𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
10097, 99sstri 3753 . . . . . . . . . . . . . 14 ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
101100a1i 11 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → ( 𝑥𝑦 𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
102 restabs 21171 . . . . . . . . . . . . 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)))
10338, 101, 78, 102syl3anc 1477 . . . . . . . . . . . 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)))
10481restin 21172 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑥𝑦 𝐵 ∈ V) → (𝐽t 𝑥𝑦 𝐵) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
10538, 44, 104syl2anc 696 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥𝑦 𝐵) = (𝐽t ( 𝑥𝑦 𝐵 𝐽)))
106103, 105eqtr4d 2797 . . . . . . . . . . 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))
107106, 40eqeltrd 2839 . . . . . . . . . 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)
108 inss1 3976 . . . . . . . . . . . . . . 15 (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑧 / 𝑥𝐵
109 ssun2 3920 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {𝑧}𝐵 ⊆ ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
110109, 39sseqtr4i 3779 . . . . . . . . . . . . . . . 16 𝑥 ∈ {𝑧}𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
11152, 110eqsstr3i 3777 . . . . . . . . . . . . . . 15 𝑧 / 𝑥𝐵 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
112108, 111sstri 3753 . . . . . . . . . . . . . 14 (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
113112a1i 11 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝑧 / 𝑥𝐵 𝐽) ⊆ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
114 restabs 21171 . . . . . . . . . . . . 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)))
11538, 113, 78, 114syl3anc 1477 . . . . . . . . . . . 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)))
11681restin 21172 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑧 / 𝑥𝐵 ∈ V) → (𝐽t 𝑧 / 𝑥𝐵) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
11738, 74, 116syl2anc 696 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑧 / 𝑥𝐵) = (𝐽t (𝑧 / 𝑥𝐵 𝐽)))
118115, 117eqtr4d 2797 . . . . . . . . . . 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))
119118, 70eqeltrd 2839 . . . . . . . . . 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)
120 eqid 2760 . . . . . . . . . . 11 (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
121120uncmp 21408 . . . . . . . . . 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)
12280, 96, 107, 119, 121syl22anc 1478 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽t 𝑥𝑦 𝐵) ∈ Comp)) → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)
123122exp32 632 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐽t 𝑥𝑦 𝐵) ∈ Comp → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
124123a2d 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)))
12537, 124syl5 34 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → ((𝑦𝐴 → (𝐽t 𝑥𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽t 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
126125a2i 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)))
127126a1i 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))))
12810, 16, 22, 28, 33, 127findcard2 8365 . . 3 (𝐴 ∈ Fin → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)))
1292, 128mpcom 38 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐴𝐴 → (𝐽t 𝑥𝐴 𝐵) ∈ Comp))
1301, 129mpi 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  csb 3674  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321   cuni 4588   ciun 4672  (class class class)co 6813  Fincfn 8121  t crest 16283  Topctop 20900  Compccmp 21391
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cmp 21392
This theorem is referenced by:  xkococnlem  21664
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