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Theorem loclly 17213
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 732 . . . . . . 7  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  j  e.  A )
2 simpl 443 . . . . . . 7  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  -> Locally  A  =  A )
31, 2eleqtrrd 2360 . . . . . 6  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  j  e. Locally  A )
4 simprr 733 . . . . . 6  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  x  e.  j )
5 llyrest 17211 . . . . . 6  |-  ( ( j  e. Locally  A  /\  x  e.  j )  ->  ( jt  x )  e. Locally  A )
63, 4, 5syl2anc 642 . . . . 5  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  (
jt  x )  e. Locally  A )
76, 2eleqtrd 2359 . . . 4  |-  ( (Locally  A  =  A  /\  ( j  e.  A  /\  x  e.  j
) )  ->  (
jt  x )  e.  A
)
87restnlly 17208 . . 3  |-  (Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
9 id 19 . . 3  |-  (Locally  A  =  A  -> Locally  A  =  A )
108, 9eqtrd 2315 . 2  |-  (Locally  A  =  A  -> 𝑛Locally  A  =  A )
11 simprl 732 . . . . . . 7  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  j  e.  A )
12 simpl 443 . . . . . . 7  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  -> 𝑛Locally  A  =  A )
1311, 12eleqtrrd 2360 . . . . . 6  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  j  e. 𝑛Locally  A
)
14 simprr 733 . . . . . 6  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  x  e.  j )
15 nllyrest 17212 . . . . . 6  |-  ( ( j  e. 𝑛Locally  A  /\  x  e.  j )  ->  (
jt  x )  e. 𝑛Locally  A )
1613, 14, 15syl2anc 642 . . . . 5  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  ( jt  x
)  e. 𝑛Locally  A )
1716, 12eleqtrd 2359 . . . 4  |-  ( (𝑛Locally  A  =  A  /\  (
j  e.  A  /\  x  e.  j )
)  ->  ( jt  x
)  e.  A )
1817restnlly 17208 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  = Locally  A
)
19 id 19 . . 3  |-  (𝑛Locally  A  =  A  -> 𝑛Locally  A  =  A )
2018, 19eqtr3d 2317 . 2  |-  (𝑛Locally  A  =  A  -> Locally  A  =  A )
2110, 20impbii 180 1  |-  (Locally  A  =  A  <-> 𝑛Locally 
A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   ↾t crest 13325  Locally clly 17190  𝑛Locally cnlly 17191
This theorem is referenced by:  topnlly  17217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-nei 16835  df-lly 17192  df-nlly 17193
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