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Theorem neitr 20894
Description: The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
Hypothesis
Ref Expression
neitr.1 𝑋 = 𝐽
Assertion
Ref Expression
neitr ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))

Proof of Theorem neitr
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . . . 6 𝑑(𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴)
2 nfv 1840 . . . . . . 7 𝑑 𝑐 (𝐽t 𝐴)
3 nfre1 2999 . . . . . . 7 𝑑𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)
42, 3nfan 1825 . . . . . 6 𝑑(𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
51, 4nfan 1825 . . . . 5 𝑑((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
6 simpl 473 . . . . . . 7 ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) → 𝑐 (𝐽t 𝐴))
76anim2i 592 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)))
8 simp-5r 808 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐 (𝐽t 𝐴))
9 simp1 1059 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ Top)
10 simp2 1060 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴𝑋)
11 neitr.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
1211restuni 20876 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
139, 10, 12syl2anc 692 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 = (𝐽t 𝐴))
1413ad5antr 769 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴 = (𝐽t 𝐴))
158, 14sseqtr4d 3621 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝐴)
1610ad5antr 769 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴𝑋)
1715, 16sstrd 3593 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝑋)
189ad5antr 769 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐽 ∈ Top)
19 simplr 791 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝐽)
2011eltopss 20637 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑒𝐽) → 𝑒𝑋)
2118, 19, 20syl2anc 692 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝑋)
2221ssdifssd 3726 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑋)
2317, 22unssd 3767 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋)
24 simpr1l 1116 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝐵𝑑)
25243anassrs 1287 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵𝑑)
26 simpr 477 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑑 = (𝑒𝐴))
2725, 26sseqtrd 3620 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵 ⊆ (𝑒𝐴))
28 inss1 3811 . . . . . . . . . 10 (𝑒𝐴) ⊆ 𝑒
2927, 28syl6ss 3595 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵𝑒)
30 inundif 4018 . . . . . . . . . 10 ((𝑒𝐴) ∪ (𝑒𝐴)) = 𝑒
31 simpr1r 1117 . . . . . . . . . . . . 13 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝑑𝑐)
32313anassrs 1287 . . . . . . . . . . . 12 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑑𝑐)
3326, 32eqsstr3d 3619 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑐)
34 unss1 3760 . . . . . . . . . . 11 ((𝑒𝐴) ⊆ 𝑐 → ((𝑒𝐴) ∪ (𝑒𝐴)) ⊆ (𝑐 ∪ (𝑒𝐴)))
3533, 34syl 17 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ((𝑒𝐴) ∪ (𝑒𝐴)) ⊆ (𝑐 ∪ (𝑒𝐴)))
3630, 35syl5eqssr 3629 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))
37 sseq2 3606 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝐵𝑏𝐵𝑒))
38 sseq1 3605 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)) ↔ 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴))))
3937, 38anbi12d 746 . . . . . . . . . 10 (𝑏 = 𝑒 → ((𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))) ↔ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))))
4039rspcev 3295 . . . . . . . . 9 ((𝑒𝐽 ∧ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
4119, 29, 36, 40syl12anc 1321 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
42 indir 3851 . . . . . . . . . . 11 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴))
43 incom 3783 . . . . . . . . . . . . 13 (𝐴 ∩ (𝑒𝐴)) = ((𝑒𝐴) ∩ 𝐴)
44 disjdif 4012 . . . . . . . . . . . . 13 (𝐴 ∩ (𝑒𝐴)) = ∅
4543, 44eqtr3i 2645 . . . . . . . . . . . 12 ((𝑒𝐴) ∩ 𝐴) = ∅
4645uneq2i 3742 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴)) = ((𝑐𝐴) ∪ ∅)
47 un0 3939 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ∅) = (𝑐𝐴)
4842, 46, 473eqtri 2647 . . . . . . . . . 10 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = (𝑐𝐴)
49 df-ss 3569 . . . . . . . . . . 11 (𝑐𝐴 ↔ (𝑐𝐴) = 𝑐)
5049biimpi 206 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴) = 𝑐)
5148, 50syl5req 2668 . . . . . . . . 9 (𝑐𝐴𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
5215, 51syl 17 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
53 vex 3189 . . . . . . . . . 10 𝑐 ∈ V
54 vex 3189 . . . . . . . . . . 11 𝑒 ∈ V
55 difexg 4768 . . . . . . . . . . 11 (𝑒 ∈ V → (𝑒𝐴) ∈ V)
5654, 55ax-mp 5 . . . . . . . . . 10 (𝑒𝐴) ∈ V
5753, 56unex 6909 . . . . . . . . 9 (𝑐 ∪ (𝑒𝐴)) ∈ V
58 sseq1 3605 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝑋 ↔ (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋))
59 sseq2 3606 . . . . . . . . . . . . 13 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑏𝑎𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
6059anbi2d 739 . . . . . . . . . . . 12 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝐵𝑏𝑏𝑎) ↔ (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
6160rexbidv 3045 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) ↔ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
6258, 61anbi12d 746 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ↔ ((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))))
63 ineq1 3785 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝐴) = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
6463eqeq2d 2631 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑐 = (𝑎𝐴) ↔ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)))
6562, 64anbi12d 746 . . . . . . . . 9 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) ↔ (((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))))
6657, 65spcev 3286 . . . . . . . 8 ((((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
6723, 41, 52, 66syl21anc 1322 . . . . . . 7 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
689ad3antrrr 765 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐽 ∈ Top)
69 uniexg 6908 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝐽 ∈ V)
709, 69syl 17 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ V)
7111, 70syl5eqel 2702 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝑋 ∈ V)
7271, 10ssexd 4765 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 ∈ V)
7372ad3antrrr 765 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐴 ∈ V)
74 simplr 791 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝑑 ∈ (𝐽t 𝐴))
75 elrest 16009 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑑 ∈ (𝐽t 𝐴) ↔ ∃𝑒𝐽 𝑑 = (𝑒𝐴)))
7675biimpa 501 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴 ∈ V) ∧ 𝑑 ∈ (𝐽t 𝐴)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7768, 73, 74, 76syl21anc 1322 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7867, 77r19.29a 3071 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
797, 78sylanl1 681 . . . . 5 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
80 simprr 795 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
815, 79, 80r19.29af 3069 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
82 inss2 3812 . . . . . . . . . 10 (𝑎𝐴) ⊆ 𝐴
83 sseq1 3605 . . . . . . . . . 10 (𝑐 = (𝑎𝐴) → (𝑐𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
8482, 83mpbiri 248 . . . . . . . . 9 (𝑐 = (𝑎𝐴) → 𝑐𝐴)
8584adantl 482 . . . . . . . 8 (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8685exlimiv 1855 . . . . . . 7 (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8786adantl 482 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐𝐴)
8813adantr 481 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝐴 = (𝐽t 𝐴))
8987, 88sseqtrd 3620 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐 (𝐽t 𝐴))
909ad4antr 767 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐽 ∈ Top)
9172ad4antr 767 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐴 ∈ V)
92 simplr 791 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝐽)
93 elrestr 16010 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏𝐽) → (𝑏𝐴) ∈ (𝐽t 𝐴))
9490, 91, 92, 93syl3anc 1323 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ∈ (𝐽t 𝐴))
95 simprl 793 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝑏)
96 simp3 1061 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝐴)
9796ad4antr 767 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝐴)
9895, 97ssind 3815 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵 ⊆ (𝑏𝐴))
99 simprr 795 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝑎)
100 ssrin 3816 . . . . . . . . . . . . . . . 16 (𝑏𝑎 → (𝑏𝐴) ⊆ (𝑎𝐴))
10199, 100syl 17 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ (𝑎𝐴))
102 simp-4r 806 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑐 = (𝑎𝐴))
103101, 102sseqtr4d 3621 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ 𝑐)
10494, 98, 103jca32 557 . . . . . . . . . . . . 13 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
105104ex 450 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) → ((𝐵𝑏𝑏𝑎) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
106105reximdva 3011 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
107106impr 648 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
108107an32s 845 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
109108expl 647 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
110109exlimdv 1858 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
111110imp 445 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
112 sseq2 3606 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝐵𝑑𝐵 ⊆ (𝑏𝐴)))
113 sseq1 3605 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝑑𝑐 ↔ (𝑏𝐴) ⊆ 𝑐))
114112, 113anbi12d 746 . . . . . . . 8 (𝑑 = (𝑏𝐴) → ((𝐵𝑑𝑑𝑐) ↔ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
115114rspcev 3295 . . . . . . 7 (((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
116115rexlimivw 3022 . . . . . 6 (∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
117111, 116syl 17 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
11889, 117jca 554 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
11981, 118impbida 876 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
120 resttop 20874 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ Top)
1219, 72, 120syl2anc 692 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝐽t 𝐴) ∈ Top)
12296, 13sseqtrd 3620 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵 (𝐽t 𝐴))
123 eqid 2621 . . . . 5 (𝐽t 𝐴) = (𝐽t 𝐴)
124123isnei 20817 . . . 4 (((𝐽t 𝐴) ∈ Top ∧ 𝐵 (𝐽t 𝐴)) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
125121, 122, 124syl2anc 692 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
126 fvex 6158 . . . . . 6 ((nei‘𝐽)‘𝐵) ∈ V
127 restval 16008 . . . . . 6 ((((nei‘𝐽)‘𝐵) ∈ V ∧ 𝐴 ∈ V) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
128126, 72, 127sylancr 694 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
129128eleq2d 2684 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ 𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))))
13096, 10sstrd 3593 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝑋)
131 eqid 2621 . . . . . . . . 9 (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) = (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))
132131elrnmpt 5332 . . . . . . . 8 (𝑐 ∈ V → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴)))
13353, 132ax-mp 5 . . . . . . 7 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴))
134 df-rex 2913 . . . . . . 7 (∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
135133, 134bitri 264 . . . . . 6 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
13611isnei 20817 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↔ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))))
137136anbi1d 740 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
138137exbidv 1847 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
139135, 138syl5bb 272 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
1409, 130, 139syl2anc 692 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
141129, 140bitrd 268 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
142119, 125, 1413bitr4d 300 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ 𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴)))
143142eqrdv 2619 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891   cuni 4402  cmpt 4673  ran crn 5075  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  neicnei 20811
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-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-nei 20812
This theorem is referenced by:  flfcntr  21757  cnextfres1  21782
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