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Theorem dfateq12d 30173
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 5140 . . . 4  |-  ( ph  ->  dom  F  =  dom  G )
41, 3eleq12d 2533 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
B  e.  dom  G
) )
51sneqd 3987 . . . . 5  |-  ( ph  ->  { A }  =  { B } )
62, 5reseq12d 5209 . . . 4  |-  ( ph  ->  ( F  |`  { A } )  =  ( G  |`  { B } ) )
76funeqd 5537 . . 3  |-  ( ph  ->  ( Fun  ( F  |`  { A } )  <->  Fun  ( G  |`  { B } ) ) )
84, 7anbi12d 710 . 2  |-  ( ph  ->  ( ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) )  <->  ( B  e.  dom  G  /\  Fun  ( G  |`  { B } ) ) ) )
9 df-dfat 30158 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
10 df-dfat 30158 . 2  |-  ( G defAt 
B  <->  ( B  e. 
dom  G  /\  Fun  ( G  |`  { B }
) ) )
118, 9, 103bitr4g 288 1  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3975   dom cdm 4938    |` cres 4940   Fun wfun 5510   defAt wdfat 30155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-res 4950  df-fun 5518  df-dfat 30158
This theorem is referenced by:  afveq12d  30177
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