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Theorem nllyrest 21099
 Description: An open subspace of an n-locally 𝐴 space is also n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyrest ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ 𝑛-Locally 𝐴)

Proof of Theorem nllyrest
Dummy variables 𝑠 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 21086 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
2 resttop 20774 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 487 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 20791 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 487 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1078 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7 simp2l 1080 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1056 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 nlly2i 21089 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
106, 7, 8, 9syl3anc 1318 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
1133ad2ant1 1075 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → (𝐽t 𝐵) ∈ Top)
12113ad2ant1 1075 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝐵) ∈ Top)
13 simp3l 1082 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐽)
14 simp3r2 1163 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝑠)
15 simp2 1055 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑥)
1615elpwid 4118 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝑥)
17 simp12r 1168 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑥𝐵)
1816, 17sstrd 3578 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝐵)
1914, 18sstrd 3578 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐵)
2063ad2ant1 1075 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐽 ∈ 𝑛-Locally 𝐴)
2120, 1syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐽 ∈ Top)
22 simp11r 1166 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵𝐽)
23 restopn2 20791 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2421, 22, 23syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2513, 19, 24mpbir2and 959 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ (𝐽t 𝐵))
26 simp3r1 1162 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑦𝑢)
27 opnneip 20733 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽t 𝐵) ∧ 𝑦𝑢) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
2812, 25, 26, 27syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
29 elssuni 4403 . . . . . . . . . . . . . . . . . 18 (𝐵𝐽𝐵 𝐽)
3022, 29syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵 𝐽)
31 eqid 2610 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
3231restuni 20776 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝐵 𝐽) → 𝐵 = (𝐽t 𝐵))
3321, 30, 32syl2anc 691 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵 = (𝐽t 𝐵))
3418, 33sseqtrd 3604 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 (𝐽t 𝐵))
35 eqid 2610 . . . . . . . . . . . . . . . 16 (𝐽t 𝐵) = (𝐽t 𝐵)
3635ssnei2 20730 . . . . . . . . . . . . . . 15 ((((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦})) ∧ (𝑢𝑠𝑠 (𝐽t 𝐵))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3712, 28, 14, 34, 36syl22anc 1319 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3837, 15elind 3760 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥))
39 restabs 20779 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑠𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
4021, 18, 22, 39syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
41 simp3r3 1164 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝑠) ∈ 𝐴)
4240, 41eqeltrd 2688 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
4338, 42jca 553 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
44433expa 1257 . . . . . . . . . . 11 (((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥) ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
4544rexlimdvaa 3014 . . . . . . . . . 10 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥) → (∃𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴) → (𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)))
4645expimpd 627 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ((𝑠 ∈ 𝒫 𝑥 ∧ ∃𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴)) → (𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)))
4746reximdv2 2997 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → (∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴) → ∃𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
4810, 47mpd 15 . . . . . . 7 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
49483expa 1257 . . . . . 6 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5049ralrimiva 2949 . . . . 5 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5150ex 449 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
525, 51sylbid 229 . . 3 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
5352ralrimiv 2948 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
54 isnlly 21082 . 2 ((𝐽t 𝐵) ∈ 𝑛-Locally 𝐴 ↔ ((𝐽t 𝐵) ∈ Top ∧ ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
553, 53, 54sylanbrc 695 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ 𝑛-Locally 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  neicnei 20711  𝑛-Locally cnlly 21078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-nei 20712  df-nlly 21080 This theorem is referenced by:  loclly  21100  nllyidm  21102
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