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Theorem loclly 17472
Description: If  A is a local property, then both Locally  A and 𝑛Locally  A simplify to  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
loclly  |-  (Locally  A  =  A  <-> 𝑛Locally 
A  =  A )

Proof of Theorem loclly
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 733 . . . . . . 7  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  j  e.  A )
2 simpl 444 . . . . . . 7  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  -> Locally  A  =  A )
31, 2eleqtrrd 2465 . . . . . 6  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  j  e. Locally  A )
4 simprr 734 . . . . . 6  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  x  e.  j )
5 llyrest 17470 . . . . . 6  |-  ( ( j  e. Locally  A  /\  x  e.  j )  ->  ( jt  x )  e. Locally  A )
63, 4, 5syl2anc 643 . . . . 5  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  (
jt  x )  e. Locally  A )
76, 2eleqtrd 2464 . . . 4  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  (
jt  x )  e.  A
)
87restnlly 17467 . . 3  |-  (Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
9 id 20 . . 3  |-  (Locally  A  =  A  -> Locally  A  =  A )
108, 9eqtrd 2420 . 2  |-  (Locally  A  =  A  -> 𝑛Locally  A  =  A )
11 simprl 733 . . . . . . 7  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  j  e.  A )
12 simpl 444 . . . . . . 7  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  -> 𝑛Locally  A  =  A )
1311, 12eleqtrrd 2465 . . . . . 6  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  j  e. 𝑛Locally  A
)
14 simprr 734 . . . . . 6  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  x  e.  j )
15 nllyrest 17471 . . . . . 6  |-  ( ( j  e. 𝑛Locally  A  /\  x  e.  j )  ->  (
jt  x )  e. 𝑛Locally  A )
1613, 14, 15syl2anc 643 . . . . 5  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  ( jt  x
)  e. 𝑛Locally  A )
1716, 12eleqtrd 2464 . . . 4  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  ( jt  x
)  e.  A )
1817restnlly 17467 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
19 id 20 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  =  A )
2018, 19eqtr3d 2422 . 2  |-  (𝑛Locally  A  =  A  -> Locally  A  =  A )
2110, 20impbii 181 1  |-  (Locally  A  =  A  <-> 𝑛Locally 
A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717  (class class class)co 6021   ↾t crest 13576  Locally clly 17449  𝑛Locally cnlly 17450
This theorem is referenced by:  topnlly  17476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-oadd 6665  df-er 6842  df-en 7047  df-fin 7050  df-fi 7352  df-rest 13578  df-topgen 13595  df-top 16887  df-bases 16889  df-topon 16890  df-nei 17086  df-lly 17451  df-nlly 17452
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