Theorem loclly 23615

Description: If 𝐴 is a local property, then both Locally 𝐴 and 𝑛-Locally 𝐴 simplify to 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
loclly	⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)

Proof of Theorem loclly

Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 782	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
2		simpl 487	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → Locally 𝐴 = 𝐴)
3	1, 2	eleqtrrd 2872	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ Locally 𝐴)
4		simprr 784	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
5		llyrest 23613	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
6	3, 4, 5	syl2anc 595	. . . . 5 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
7	6, 2	eleqtrd 2871	. . . 4 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
8	7	restnlly 23610	. . 3 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9		id 23	. . 3 ⊢ (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
10	8, 9	eqtrd 2804	. 2 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11		simprl 782	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
12		simpl 487	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
13	11, 12	eleqtrrd 2872	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14		simprr 784	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
15		nllyrest 23614	. . . . . 6 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
16	13, 14, 15	syl2anc 595	. . . . 5 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
17	16, 12	eleqtrd 2871	. . . 4 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
18	17	restnlly 23610	. . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
19		id 23	. . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
20	18, 19	eqtr3d 2806	. 2 ⊢ (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
21	10, 20	impbii 212	1 ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  (class class class)co 7413   ↾_t crest 17475  Locally clly 23592  𝑛-Locally cnlly 23593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-en 8946  df-fin 8949  df-fi 9373  df-rest 17477  df-topgen 17498  df-top 23022  df-topon 23039  df-bases 23074  df-nei 23226  df-lly 23594  df-nlly 23595

This theorem is referenced by:  topnlly  23619