Theorem loclly 23806

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 783	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
2		simpl 488	. . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → Locally 𝐴 = 𝐴)
3	1, 2	eleqtrrd 2863	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ Locally 𝐴)
4		simprr 785	. . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
5		llyrest 23804	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
6	3, 4, 5	syl2anc 596	. . . . 5 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
7	6, 2	eleqtrd 2862	. . . 4 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
8	7	restnlly 23801	. . 3 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9		id 23	. . 3 ⊢ (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
10	8, 9	eqtrd 2795	. 2 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11		simprl 783	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴)
12		simpl 488	. . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
13	11, 12	eleqtrrd 2863	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14		simprr 785	. . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗)
15		nllyrest 23805	. . . . . 6 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
16	13, 14, 15	syl2anc 596	. . . . 5 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
17	16, 12	eleqtrd 2862	. . . 4 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
18	17	restnlly 23801	. . 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 2797	. 2 ⊢ (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
21	10, 20	impbii 212	1 ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  (class class class)co 7420   ↾_t crest 17591  Locally clly 23783  𝑛-Locally cnlly 23784

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-en 8974  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-nei 23416  df-lly 23785  df-nlly 23786

This theorem is used by:  topnlly  23810