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Theorem loclly 17542
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 2512 . . . . . 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 17540 . . . . . 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 2511 . . . 4  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  (
jt  x )  e.  A
)
87restnlly 17537 . . 3  |-  (Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
9 id 20 . . 3  |-  (Locally  A  =  A  -> Locally  A  =  A )
108, 9eqtrd 2467 . 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 2512 . . . . . 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 17541 . . . . . 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 2511 . . . 4  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  ( jt  x
)  e.  A )
1817restnlly 17537 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
19 id 20 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  =  A )
2018, 19eqtr3d 2469 . 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 1652    e. wcel 1725  (class class class)co 6073   ↾t crest 13640  Locally clly 17519  𝑛Locally cnlly 17520
This theorem is referenced by:  topnlly  17546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-nei 17154  df-lly 17521  df-nlly 17522
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