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Theorem dfateq12d 28097
Description: Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
dfateq12d.1  |-  ( ph  ->  F  =  G )
dfateq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
dfateq12d  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )

Proof of Theorem dfateq12d
StepHypRef Expression
1 dfateq12d.2 . . . 4  |-  ( ph  ->  A  =  B )
2 dfateq12d.1 . . . . 5  |-  ( ph  ->  F  =  G )
32dmeqd 4897 . . . 4  |-  ( ph  ->  dom  F  =  dom  G )
41, 3eleq12d 2364 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
B  e.  dom  G
) )
51sneqd 3666 . . . . 5  |-  ( ph  ->  { A }  =  { B } )
62, 5reseq12d 4972 . . . 4  |-  ( ph  ->  ( F  |`  { A } )  =  ( G  |`  { B } ) )
76funeqd 5292 . . 3  |-  ( ph  ->  ( Fun  ( F  |`  { A } )  <->  Fun  ( G  |`  { B } ) ) )
84, 7anbi12d 691 . 2  |-  ( ph  ->  ( ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) )  <->  ( B  e.  dom  G  /\  Fun  ( G  |`  { B } ) ) ) )
9 df-dfat 28077 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
10 df-dfat 28077 . 2  |-  ( G defAt 
B  <->  ( B  e. 
dom  G  /\  Fun  ( G  |`  { B }
) ) )
118, 9, 103bitr4g 279 1  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   dom cdm 4705    |` cres 4707   Fun wfun 5265   defAt wdfat 28074
This theorem is referenced by:  afveq12d  28101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-fun 5273  df-dfat 28077
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