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Theorem dfateq12d 37595
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 5028 . . . 4  |-  ( ph  ->  dom  F  =  dom  G )
41, 3eleq12d 2486 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
B  e.  dom  G
) )
51sneqd 3986 . . . . 5  |-  ( ph  ->  { A }  =  { B } )
62, 5reseq12d 5097 . . . 4  |-  ( ph  ->  ( F  |`  { A } )  =  ( G  |`  { B } ) )
76funeqd 5592 . . 3  |-  ( ph  ->  ( Fun  ( F  |`  { A } )  <->  Fun  ( G  |`  { B } ) ) )
84, 7anbi12d 711 . 2  |-  ( ph  ->  ( ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) )  <->  ( B  e.  dom  G  /\  Fun  ( G  |`  { B } ) ) ) )
9 df-dfat 37582 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
10 df-dfat 37582 . 2  |-  ( G defAt 
B  <->  ( B  e. 
dom  G  /\  Fun  ( G  |`  { B }
) ) )
118, 9, 103bitr4g 290 1  |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   {csn 3974   dom cdm 4825    |` cres 4827   Fun wfun 5565   defAt wdfat 37579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-res 4837  df-fun 5573  df-dfat 37582
This theorem is referenced by:  afveq12d  37599
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