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Theorem neitr 21264
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 2009 . . . . . 6 𝑑(𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴)
2 nfv 2009 . . . . . . 7 𝑑 𝑐 (𝐽t 𝐴)
3 nfre1 3151 . . . . . . 7 𝑑𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)
42, 3nfan 1998 . . . . . 6 𝑑(𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
51, 4nfan 1998 . . . . 5 𝑑((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
6 simpl 474 . . . . . . 7 ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) → 𝑐 (𝐽t 𝐴))
76anim2i 610 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)))
8 simp-5r 807 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐 (𝐽t 𝐴))
9 simp1 1166 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ Top)
10 simp2 1167 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴𝑋)
11 neitr.1 . . . . . . . . . . . . . 14 𝑋 = 𝐽
1211restuni 21246 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
139, 10, 12syl2anc 579 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 = (𝐽t 𝐴))
1413ad5antr 728 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴 = (𝐽t 𝐴))
158, 14sseqtr4d 3802 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝐴)
1610ad5antr 728 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐴𝑋)
1715, 16sstrd 3771 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑐𝑋)
189ad5antr 728 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐽 ∈ Top)
19 simplr 785 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝐽)
2011eltopss 20991 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑒𝐽) → 𝑒𝑋)
2118, 19, 20syl2anc 579 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒𝑋)
2221ssdifssd 3910 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑋)
2317, 22unssd 3951 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋)
24 simpr1l 1305 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝐵𝑑)
25243anassrs 1469 . . . . . . . . . . 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 3801 . . . . . . . . . 10 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵 ⊆ (𝑒𝐴))
28 inss1 3992 . . . . . . . . . 10 (𝑒𝐴) ⊆ 𝑒
2927, 28syl6ss 3773 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝐵𝑒)
30 inundif 4206 . . . . . . . . . 10 ((𝑒𝐴) ∪ (𝑒𝐴)) = 𝑒
31 simpr1r 1307 . . . . . . . . . . . . 13 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ ((𝐵𝑑𝑑𝑐) ∧ 𝑒𝐽𝑑 = (𝑒𝐴))) → 𝑑𝑐)
32313anassrs 1469 . . . . . . . . . . . 12 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑑𝑐)
3326, 32eqsstr3d 3800 . . . . . . . . . . 11 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → (𝑒𝐴) ⊆ 𝑐)
34 unss1 3944 . . . . . . . . . . 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 3810 . . . . . . . . 9 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))
37 sseq2 3787 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝐵𝑏𝐵𝑒))
38 sseq1 3786 . . . . . . . . . . 11 (𝑏 = 𝑒 → (𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)) ↔ 𝑒 ⊆ (𝑐 ∪ (𝑒𝐴))))
3937, 38anbi12d 624 . . . . . . . . . 10 (𝑏 = 𝑒 → ((𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))) ↔ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))))
4039rspcev 3461 . . . . . . . . 9 ((𝑒𝐽 ∧ (𝐵𝑒𝑒 ⊆ (𝑐 ∪ (𝑒𝐴)))) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
4119, 29, 36, 40syl12anc 865 . . . . . . . 8 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
42 indir 4040 . . . . . . . . . . 11 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴))
43 incom 3967 . . . . . . . . . . . . 13 (𝐴 ∩ (𝑒𝐴)) = ((𝑒𝐴) ∩ 𝐴)
44 disjdif 4200 . . . . . . . . . . . . 13 (𝐴 ∩ (𝑒𝐴)) = ∅
4543, 44eqtr3i 2789 . . . . . . . . . . . 12 ((𝑒𝐴) ∩ 𝐴) = ∅
4645uneq2i 3926 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ((𝑒𝐴) ∩ 𝐴)) = ((𝑐𝐴) ∪ ∅)
47 un0 4129 . . . . . . . . . . 11 ((𝑐𝐴) ∪ ∅) = (𝑐𝐴)
4842, 46, 473eqtri 2791 . . . . . . . . . 10 ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴) = (𝑐𝐴)
49 df-ss 3746 . . . . . . . . . . 11 (𝑐𝐴 ↔ (𝑐𝐴) = 𝑐)
5049biimpi 207 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴) = 𝑐)
5148, 50syl5req 2812 . . . . . . . . 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 3353 . . . . . . . . . 10 𝑐 ∈ V
54 vex 3353 . . . . . . . . . . 11 𝑒 ∈ V
55 difexg 4969 . . . . . . . . . . 11 (𝑒 ∈ V → (𝑒𝐴) ∈ V)
5654, 55ax-mp 5 . . . . . . . . . 10 (𝑒𝐴) ∈ V
5753, 56unex 7154 . . . . . . . . 9 (𝑐 ∪ (𝑒𝐴)) ∈ V
58 sseq1 3786 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝑋 ↔ (𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋))
59 sseq2 3787 . . . . . . . . . . . . 13 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑏𝑎𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))
6059anbi2d 622 . . . . . . . . . . . 12 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝐵𝑏𝑏𝑎) ↔ (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
6160rexbidv 3199 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) ↔ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))))
6258, 61anbi12d 624 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ↔ ((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴))))))
63 ineq1 3969 . . . . . . . . . . 11 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑎𝐴) = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))
6463eqeq2d 2775 . . . . . . . . . 10 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (𝑐 = (𝑎𝐴) ↔ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)))
6562, 64anbi12d 624 . . . . . . . . 9 (𝑎 = (𝑐 ∪ (𝑒𝐴)) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) ↔ (((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴))))
6657, 65spcev 3452 . . . . . . . 8 ((((𝑐 ∪ (𝑒𝐴)) ⊆ 𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏 ⊆ (𝑐 ∪ (𝑒𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒𝐴)) ∩ 𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
6723, 41, 52, 66syl21anc 866 . . . . . . 7 (((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) ∧ 𝑒𝐽) ∧ 𝑑 = (𝑒𝐴)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
689ad3antrrr 721 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐽 ∈ Top)
69 uniexg 7153 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝐽 ∈ V)
709, 69syl 17 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐽 ∈ V)
7111, 70syl5eqel 2848 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝑋 ∈ V)
7271, 10ssexd 4966 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐴 ∈ V)
7372ad3antrrr 721 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝐴 ∈ V)
74 simplr 785 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → 𝑑 ∈ (𝐽t 𝐴))
75 elrest 16356 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑑 ∈ (𝐽t 𝐴) ↔ ∃𝑒𝐽 𝑑 = (𝑒𝐴)))
7675biimpa 468 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴 ∈ V) ∧ 𝑑 ∈ (𝐽t 𝐴)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7768, 73, 74, 76syl21anc 866 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑒𝐽 𝑑 = (𝑒𝐴))
7867, 77r19.29a 3225 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 (𝐽t 𝐴)) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
797, 78sylanl1 670 . . . . 5 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) ∧ 𝑑 ∈ (𝐽t 𝐴)) ∧ (𝐵𝑑𝑑𝑐)) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
80 simprr 789 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
815, 79, 80r19.29af 3223 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))) → ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)))
82 inss2 3993 . . . . . . . . . 10 (𝑎𝐴) ⊆ 𝐴
83 sseq1 3786 . . . . . . . . . 10 (𝑐 = (𝑎𝐴) → (𝑐𝐴 ↔ (𝑎𝐴) ⊆ 𝐴))
8482, 83mpbiri 249 . . . . . . . . 9 (𝑐 = (𝑎𝐴) → 𝑐𝐴)
8584adantl 473 . . . . . . . 8 (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8685exlimiv 2025 . . . . . . 7 (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → 𝑐𝐴)
8786adantl 473 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐𝐴)
8813adantr 472 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝐴 = (𝐽t 𝐴))
8987, 88sseqtrd 3801 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → 𝑐 (𝐽t 𝐴))
909ad4antr 724 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐽 ∈ Top)
9172ad4antr 724 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐴 ∈ V)
92 simplr 785 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝐽)
93 elrestr 16357 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏𝐽) → (𝑏𝐴) ∈ (𝐽t 𝐴))
9490, 91, 92, 93syl3anc 1490 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ∈ (𝐽t 𝐴))
95 simprl 787 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝑏)
96 simp3 1168 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝐴)
9796ad4antr 724 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵𝐴)
9895, 97ssind 3996 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝐵 ⊆ (𝑏𝐴))
99 simprr 789 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑏𝑎)
10099ssrind 3999 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ (𝑎𝐴))
101 simp-4r 803 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → 𝑐 = (𝑎𝐴))
102100, 101sseqtr4d 3802 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → (𝑏𝐴) ⊆ 𝑐)
10394, 98, 102jca32 511 . . . . . . . . . . . . 13 ((((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) ∧ (𝐵𝑏𝑏𝑎)) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
104103ex 401 . . . . . . . . . . . 12 (((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) ∧ 𝑏𝐽) → ((𝐵𝑏𝑏𝑎) → ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
105104reximdva 3163 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ 𝑎𝑋) → (∃𝑏𝐽 (𝐵𝑏𝑏𝑎) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
106105impr 446 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ 𝑐 = (𝑎𝐴)) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
107106an32s 642 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
108107expl 449 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
109108exlimdv 2028 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴)) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐))))
110109imp 395 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
111 sseq2 3787 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝐵𝑑𝐵 ⊆ (𝑏𝐴)))
112 sseq1 3786 . . . . . . . . 9 (𝑑 = (𝑏𝐴) → (𝑑𝑐 ↔ (𝑏𝐴) ⊆ 𝑐))
113111, 112anbi12d 624 . . . . . . . 8 (𝑑 = (𝑏𝐴) → ((𝐵𝑑𝑑𝑐) ↔ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)))
114113rspcev 3461 . . . . . . 7 (((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
115114rexlimivw 3176 . . . . . 6 (∃𝑏𝐽 ((𝑏𝐴) ∈ (𝐽t 𝐴) ∧ (𝐵 ⊆ (𝑏𝐴) ∧ (𝑏𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
116110, 115syl 17 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))
11789, 116jca 507 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) ∧ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))) → (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)))
11881, 117impbida 835 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
119 resttop 21244 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ Top)
1209, 72, 119syl2anc 579 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝐽t 𝐴) ∈ Top)
12196, 13sseqtrd 3801 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵 (𝐽t 𝐴))
122 eqid 2765 . . . . 5 (𝐽t 𝐴) = (𝐽t 𝐴)
123122isnei 21187 . . . 4 (((𝐽t 𝐴) ∈ Top ∧ 𝐵 (𝐽t 𝐴)) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
124120, 121, 123syl2anc 579 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ (𝑐 (𝐽t 𝐴) ∧ ∃𝑑 ∈ (𝐽t 𝐴)(𝐵𝑑𝑑𝑐))))
125 fvex 6388 . . . . . 6 ((nei‘𝐽)‘𝐵) ∈ V
126 restval 16355 . . . . . 6 ((((nei‘𝐽)‘𝐵) ∈ V ∧ 𝐴 ∈ V) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
127125, 72, 126sylancr 581 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)))
128127eleq2d 2830 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ 𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))))
12996, 10sstrd 3771 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → 𝐵𝑋)
130 eqid 2765 . . . . . . . . 9 (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) = (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴))
131130elrnmpt 5541 . . . . . . . 8 (𝑐 ∈ V → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴)))
13253, 131ax-mp 5 . . . . . . 7 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴))
133 df-rex 3061 . . . . . . 7 (∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎𝐴) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
134132, 133bitri 266 . . . . . 6 (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)))
13511isnei 21187 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↔ (𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎))))
136135anbi1d 623 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
137136exbidv 2016 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
138134, 137syl5bb 274 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
1399, 129, 138syl2anc 579 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎𝐴)) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
140128, 139bitrd 270 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ ∃𝑎((𝑎𝑋 ∧ ∃𝑏𝐽 (𝐵𝑏𝑏𝑎)) ∧ 𝑐 = (𝑎𝐴))))
141118, 124, 1403bitr4d 302 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → (𝑐 ∈ ((nei‘(𝐽t 𝐴))‘𝐵) ↔ 𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴)))
142141eqrdv 2763 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079   cuni 4594  cmpt 4888  ran crn 5278  cfv 6068  (class class class)co 6842  t crest 16349  Topctop 20977  neicnei 21181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-en 8161  df-fin 8164  df-fi 8524  df-rest 16351  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-nei 21182
This theorem is referenced by:  flfcntr  22126  cnextfres1  22151
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