Theorem loclly 23452

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 771	. . . . . . 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 2839	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ Locally 𝐴)
4		simprr 773	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
5		llyrest 23450	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
6	3, 4, 5	syl2anc 585	. . . . 5 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
7	6, 2	eleqtrd 2838	. . . 4 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
8	7	restnlly 23447	. . 3 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9		id 22	. . 3 ⊢ (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
10	8, 9	eqtrd 2771	. 2 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11		simprl 771	. . . . . . 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 2839	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14		simprr 773	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
15		nllyrest 23451	. . . . . 6 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
16	13, 14, 15	syl2anc 585	. . . . 5 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
17	16, 12	eleqtrd 2838	. . . 4 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
18	17	restnlly 23447	. . 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 2773	. 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 1542   ∈ wcel 2114  (class class class)co 7367   ↾_t crest 17383  Locally clly 23429  𝑛-Locally cnlly 23430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-nei 23063  df-lly 23431  df-nlly 23432

This theorem is referenced by:  topnlly  23456