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

Theorem nllyrest 21510
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 21497 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
2 resttop 21185 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 569 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 21202 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 569 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1239 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7 simp2l 1241 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1132 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 nlly2i 21500 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
106, 7, 8, 9syl3anc 1476 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑠 ∈ 𝒫 𝑥𝑢𝐽 (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
1133ad2ant1 1127 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → (𝐽t 𝐵) ∈ Top)
12113ad2ant1 1127 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝐵) ∈ Top)
13 simp3l 1243 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐽)
14 simp3r2 1366 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝑠)
15 simp2 1131 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑥)
1615elpwid 4309 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝑥)
17 simp12r 1371 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑥𝐵)
1816, 17sstrd 3762 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠𝐵)
1914, 18sstrd 3762 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢𝐵)
2063ad2ant1 1127 . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵𝐽)
23 restopn2 21202 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2421, 22, 23syl2anc 573 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝑢 ∈ (𝐽t 𝐵) ↔ (𝑢𝐽𝑢𝐵)))
2513, 19, 24mpbir2and 692 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ (𝐽t 𝐵))
26 simp3r1 1365 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑦𝑢)
27 opnneip 21144 . . . . . . . . . . . . . . . 16 (((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽t 𝐵) ∧ 𝑦𝑢) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
2812, 25, 26, 27syl3anc 1476 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
29 elssuni 4603 . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
3231restuni 21187 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝐵 𝐽) → 𝐵 = (𝐽t 𝐵))
3321, 30, 32syl2anc 573 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝐵 = (𝐽t 𝐵))
3418, 33sseqtrd 3790 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 (𝐽t 𝐵))
35 eqid 2771 . . . . . . . . . . . . . . . 16 (𝐽t 𝐵) = (𝐽t 𝐵)
3635ssnei2 21141 . . . . . . . . . . . . . . 15 ((((𝐽t 𝐵) ∈ Top ∧ 𝑢 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦})) ∧ (𝑢𝑠𝑠 (𝐽t 𝐵))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3712, 28, 14, 34, 36syl22anc 1477 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ ((nei‘(𝐽t 𝐵))‘{𝑦}))
3837, 15elind 3949 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → 𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥))
39 restabs 21190 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑠𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
4021, 18, 22, 39syl3anc 1476 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) = (𝐽t 𝑠))
41 simp3r3 1367 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → (𝐽t 𝑠) ∈ 𝐴)
4240, 41eqeltrd 2850 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢𝐽 ∧ (𝑦𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
4338, 42jca 501 . . . . . . . . . . . 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 1111 . . . . . . . . . . 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 3180 . . . . . . . . . 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 441 . . . . . . . . 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 3162 . . . . . . . 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 1111 . . . . . 6 ((((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5049ralrimiva 3115 . . . . 5 (((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
5150ex 397 . . . 4 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
525, 51sylbid 230 . . 3 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴))
5352ralrimiv 3114 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑠 ∈ (((nei‘(𝐽t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐵) ↾t 𝑠) ∈ 𝐴)
54 isnlly 21493 . 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 572 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4297  {csn 4316   cuni 4574  cfv 6031  (class class class)co 6793  t crest 16289  Topctop 20918  neicnei 21122  𝑛-Locally cnlly 21489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-nei 21123  df-nlly 21491
This theorem is referenced by:  loclly  21511  nllyidm  21513
  Copyright terms: Public domain W3C validator