MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nllyrest Structured version   Visualization version   GIF version

Theorem nllyrest 22383
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 22370 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
2 resttop 22057 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 583 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 22074 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 583 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1199 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7 simp2l 1201 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1140 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 nlly2i 22373 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
106, 7, 8, 9syl3anc 1373 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
1133ad2ant1 1135 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → (𝐽t 𝐵) ∈ Top)
12113ad2ant1 1135 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝐵) ∈ Top)
13 simp3l 1203 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐽)
14 simp3r2 1284 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝑠)
15 simp2 1139 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑥)
1615elpwid 4524 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝑥)
17 simp12r 1289 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑥𝐵)
1816, 17sstrd 3911 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝐵)
1914, 18sstrd 3911 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐵)
2063ad2ant1 1135 . . . . . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵𝐽)
23 restopn2 22074 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2421, 22, 23syl2anc 587 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2513, 19, 24mpbir2and 713 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ (𝐽t 𝐵))
26 simp3r1 1283 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑦𝑢)
27 opnneip 22016 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽t 𝐵) ∧ 𝑦𝑢) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
2812, 25, 26, 27syl3anc 1373 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
29 elssuni 4851 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
3231restuni 22059 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝐵 𝐽) → 𝐵 = (𝐽t 𝐵))
3321, 30, 32syl2anc 587 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵 = (𝐽t 𝐵))
3418, 33sseqtrd 3941 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 (𝐽t 𝐵))
35 eqid 2737 . . . . . . . . . . . . . . . 16 (𝐽t 𝐵) = (𝐽t 𝐵)
3635ssnei2 22013 . . . . . . . . . . . . . . 15 ((((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦})) ∧ (𝑢𝑠𝑠 (𝐽t 𝐵))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3712, 28, 14, 34, 36syl22anc 839 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3837, 15elind 4108 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥))
39 restabs 22062 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑠𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
4021, 18, 22, 39syl3anc 1373 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
41 simp3r3 1285 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝑠) ∈ 𝐴)
4240, 41eqeltrd 2838 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
4338, 42jca 515 . . . . . . . . . . . 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 1120 . . . . . . . . . . 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 3204 . . . . . . . . . 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 457 . . . . . . . . 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 3190 . . . . . . . 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 1120 . . . . . 6 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5049ralrimiva 3105 . . . . 5 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5150ex 416 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
525, 51sylbid 243 . . 3 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
5352ralrimiv 3104 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
54 isnlly 22366 . 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 586 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866  𝒫 cpw 4513  {csn 4541   cuni 4819  cfv 6380  (class class class)co 7213  t crest 16925  Topctop 21790  neicnei 21994  𝑛-Locally cnlly 22362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-en 8627  df-fin 8630  df-fi 9027  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-nei 21995  df-nlly 22364
This theorem is referenced by:  loclly  22384  nllyidm  22386
  Copyright terms: Public domain W3C validator