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Theorem neitr 23188
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 1914 . . . . . 6 𝑑(𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴)
2 nfv 1914 . . . . . . 7 𝑑 𝑐 (𝐽t 𝐴)
3 nfre1 3285 . . . . . . 7 𝑑𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)
42, 3nfan 1899 . . . . . 6 𝑑(𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
51, 4nfan 1899 . . . . 5 𝑑((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
6 simpl 482 . . . . . . 7 ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) → 𝑐 (𝐽t 𝐴))
76anim2i 617 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)))
8 simp-5r 786 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐 (𝐽t 𝐴))
9 simp1 1137 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ Top)
10 simp2 1138 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴𝑋)
11 neitr.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
1211restuni 23170 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
139, 10, 12syl2anc 584 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 = (𝐽t 𝐴))
1413ad5antr 734 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴 = (𝐽t 𝐴))
158, 14sseqtrrd 4021 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝐴)
1610ad5antr 734 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴𝑋)
1715, 16sstrd 3994 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝑋)
189ad5antr 734 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐽 ∈ Top)
19 simplr 769 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝐽)
2011eltopss 22913 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑒𝐽) → 𝑒𝑋)
2118, 19, 20syl2anc 584 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝑋)
2221ssdifssd 4147 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑋)
2317, 22unssd 4192 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋)
24 simpr1l 1231 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝐵𝑑)
25243anassrs 1361 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵𝑑)
26 simpr 484 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑑 = (𝑒𝐴))
2725, 26sseqtrd 4020 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵 ⊆ (𝑒𝐴))
28 inss1 4237 . . . . . . . . . 10 (𝑒𝐴) ⊆ 𝑒
2927, 28sstrdi 3996 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵𝑒)
30 inundif 4479 . . . . . . . . . 10 ((𝑒𝐴) ∪ (𝑒𝐴)) = 𝑒
31 simpr1r 1232 . . . . . . . . . . . . 13 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝑑𝑐)
32313anassrs 1361 . . . . . . . . . . . 12 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑑𝑐)
3326, 32eqsstrrd 4019 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑐)
34 unss1 4185 . . . . . . . . . . 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, 35eqsstrrid 4023 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))
37 sseq2 4010 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝐵𝑏𝐵𝑒))
38 sseq1 4009 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)) ↔ 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴))))
3937, 38anbi12d 632 . . . . . . . . . 10 (𝑏 = 𝑒 → ((𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))) ↔ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))))
4039rspcev 3622 . . . . . . . . 9 ((𝑒𝐽 ∧ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
4119, 29, 36, 40syl12anc 837 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
42 indir 4286 . . . . . . . . . . 11 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴))
43 disjdifr 4473 . . . . . . . . . . . 12 ((𝑒𝐴) ∩ 𝐴) = ∅
4443uneq2i 4165 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴)) = ((𝑐𝐴) ∪ ∅)
45 un0 4394 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ∅) = (𝑐𝐴)
4642, 44, 453eqtri 2769 . . . . . . . . . 10 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = (𝑐𝐴)
47 dfss2 3969 . . . . . . . . . . 11 (𝑐𝐴 ↔ (𝑐𝐴) = 𝑐)
4847biimpi 216 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴) = 𝑐)
4946, 48eqtr2id 2790 . . . . . . . . 9 (𝑐𝐴𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
5015, 49syl 17 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
51 vex 3484 . . . . . . . . . 10 𝑐 ∈ V
52 vex 3484 . . . . . . . . . . 11 𝑒 ∈ V
5352difexi 5330 . . . . . . . . . 10 (𝑒𝐴) ∈ V
5451, 53unex 7764 . . . . . . . . 9 (𝑐 ∪ (𝑒𝐴)) ∈ V
55 sseq1 4009 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝑋 ↔ (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋))
56 sseq2 4010 . . . . . . . . . . . . 13 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑏𝑎𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
5756anbi2d 630 . . . . . . . . . . . 12 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝐵𝑏𝑏𝑎) ↔ (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
5857rexbidv 3179 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) ↔ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
5955, 58anbi12d 632 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ↔ ((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))))
60 ineq1 4213 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝐴) = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
6160eqeq2d 2748 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑐 = (𝑎𝐴) ↔ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)))
6259, 61anbi12d 632 . . . . . . . . 9 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) ↔ (((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))))
6354, 62spcev 3606 . . . . . . . 8 ((((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
6423, 41, 50, 63syl21anc 838 . . . . . . 7 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
659ad3antrrr 730 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐽 ∈ Top)
669uniexd 7762 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ V)
6711, 66eqeltrid 2845 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝑋 ∈ V)
6867, 10ssexd 5324 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 ∈ V)
6968ad3antrrr 730 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐴 ∈ V)
70 simplr 769 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝑑 ∈ (𝐽t 𝐴))
71 elrest 17472 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑑 ∈ (𝐽t 𝐴) ↔ ∃𝑒𝐽 𝑑 = (𝑒𝐴)))
7271biimpa 476 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴 ∈ V) ∧ 𝑑 ∈ (𝐽t 𝐴)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7365, 69, 70, 72syl21anc 838 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7464, 73r19.29a 3162 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
757, 74sylanl1 680 . . . . 5 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
76 simprr 773 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
775, 75, 76r19.29af 3268 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
78 inss2 4238 . . . . . . . . . 10 (𝑎𝐴) ⊆ 𝐴
79 sseq1 4009 . . . . . . . . . 10 (𝑐 = (𝑎𝐴) → (𝑐𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
8078, 79mpbiri 258 . . . . . . . . 9 (𝑐 = (𝑎𝐴) → 𝑐𝐴)
8180adantl 481 . . . . . . . 8 (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8281exlimiv 1930 . . . . . . 7 (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8382adantl 481 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐𝐴)
8413adantr 480 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝐴 = (𝐽t 𝐴))
8583, 84sseqtrd 4020 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐 (𝐽t 𝐴))
869ad4antr 732 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐽 ∈ Top)
8768ad4antr 732 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐴 ∈ V)
88 simplr 769 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝐽)
89 elrestr 17473 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏𝐽) → (𝑏𝐴) ∈ (𝐽t 𝐴))
9086, 87, 88, 89syl3anc 1373 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ∈ (𝐽t 𝐴))
91 simprl 771 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝑏)
92 simp3 1139 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝐴)
9392ad4antr 732 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝐴)
9491, 93ssind 4241 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵 ⊆ (𝑏𝐴))
95 simprr 773 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝑎)
9695ssrind 4244 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ (𝑎𝐴))
97 simp-4r 784 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑐 = (𝑎𝐴))
9896, 97sseqtrrd 4021 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ 𝑐)
9990, 94, 98jca32 515 . . . . . . . . . . . . 13 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
10099ex 412 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) → ((𝐵𝑏𝑏𝑎) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
101100reximdva 3168 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
102101impr 454 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
103102an32s 652 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
104103expl 457 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
105104exlimdv 1933 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
106105imp 406 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
107 sseq2 4010 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝐵𝑑𝐵 ⊆ (𝑏𝐴)))
108 sseq1 4009 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝑑𝑐 ↔ (𝑏𝐴) ⊆ 𝑐))
109107, 108anbi12d 632 . . . . . . . 8 (𝑑 = (𝑏𝐴) → ((𝐵𝑑𝑑𝑐) ↔ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
110109rspcev 3622 . . . . . . 7 (((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
111110rexlimivw 3151 . . . . . 6 (∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
112106, 111syl 17 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
11385, 112jca 511 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
11477, 113impbida 801 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
115 resttop 23168 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ Top)
1169, 68, 115syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝐽t 𝐴) ∈ Top)
11792, 13sseqtrd 4020 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵 (𝐽t 𝐴))
118 eqid 2737 . . . . 5 (𝐽t 𝐴) = (𝐽t 𝐴)
119118isnei 23111 . . . 4 (((𝐽t 𝐴) ∈ Top ∧ 𝐵 (𝐽t 𝐴)) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
120116, 117, 119syl2anc 584 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
121 fvex 6919 . . . . . 6 ((nei‘𝐽)‘𝐵) ∈ V
122 restval 17471 . . . . . 6 ((((nei‘𝐽)‘𝐵) ∈ V ∧ 𝐴 ∈ V) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
123121, 68, 122sylancr 587 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
124123eleq2d 2827 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ 𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))))
12592, 10sstrd 3994 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝑋)
126 eqid 2737 . . . . . . . . 9 (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) = (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))
127126elrnmpt 5969 . . . . . . . 8 (𝑐 ∈ V → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴)))
128127elv 3485 . . . . . . 7 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴))
129 df-rex 3071 . . . . . . 7 (∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
130128, 129bitri 275 . . . . . 6 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
13111isnei 23111 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↔ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))))
132131anbi1d 631 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
133132exbidv 1921 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
134130, 133bitrid 283 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
1359, 125, 134syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
136124, 135bitrd 279 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
137114, 120, 1363bitr4d 311 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ 𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴)))
138137eqrdv 2735 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333   cuni 4907  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-nei 23106
This theorem is referenced by:  flfcntr  24051  cnextfres1  24076
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