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Theorem dfateq12d 27655
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 5005 . . . 4  |-  ( ph  ->  dom  F  =  dom  G )
41, 3eleq12d 2448 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
B  e.  dom  G
) )
51sneqd 3763 . . . . 5  |-  ( ph  ->  { A }  =  { B } )
62, 5reseq12d 5080 . . . 4  |-  ( ph  ->  ( F  |`  { A } )  =  ( G  |`  { B } ) )
76funeqd 5408 . . 3  |-  ( ph  ->  ( Fun  ( F  |`  { A } )  <->  Fun  ( G  |`  { B } ) ) )
84, 7anbi12d 692 . 2  |-  ( ph  ->  ( ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) )  <->  ( B  e.  dom  G  /\  Fun  ( G  |`  { B } ) ) ) )
9 df-dfat 27635 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
10 df-dfat 27635 . 2  |-  ( G defAt 
B  <->  ( B  e. 
dom  G  /\  Fun  ( G  |`  { B }
) ) )
118, 9, 103bitr4g 280 1  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3750   dom cdm 4811    |` cres 4813   Fun wfun 5381   defAt wdfat 27632
This theorem is referenced by:  afveq12d  27659
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-fun 5389  df-dfat 27635
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