Theorem loclly 23423

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 770	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
2		simpl 482	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → Locally 𝐴 = 𝐴)
3	1, 2	eleqtrrd 2837	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ Locally 𝐴)
4		simprr 772	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
5		llyrest 23421	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
6	3, 4, 5	syl2anc 584	. . . . 5 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
7	6, 2	eleqtrd 2836	. . . 4 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
8	7	restnlly 23418	. . 3 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9		id 22	. . 3 ⊢ (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
10	8, 9	eqtrd 2770	. 2 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11		simprl 770	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
12		simpl 482	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
13	11, 12	eleqtrrd 2837	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14		simprr 772	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
15		nllyrest 23422	. . . . . 6 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
16	13, 14, 15	syl2anc 584	. . . . 5 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
17	16, 12	eleqtrd 2836	. . . 4 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
18	17	restnlly 23418	. . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
19		id 22	. . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
20	18, 19	eqtr3d 2772	. 2 ⊢ (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
21	10, 20	impbii 209	1 ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  (class class class)co 7403   ↾_t crest 17432  Locally clly 23400  𝑛-Locally cnlly 23401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-en 8958  df-fin 8961  df-fi 9421  df-rest 17434  df-topgen 17455  df-top 22830  df-topon 22847  df-bases 22882  df-nei 23034  df-lly 23402  df-nlly 23403

This theorem is referenced by:  topnlly  23427