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Theorem feq123 5586
Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
Assertion
Ref Expression
feq123  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )

Proof of Theorem feq123
StepHypRef Expression
1 simp1 958 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  F  =  G )
2 simp2 959 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  A  =  C )
3 simp3 960 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  B  =  D )
41, 2, 3feq123d 5585 1  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653   -->wf 5452
This theorem is referenced by:  mbfresfi  26255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-fun 5458  df-fn 5459  df-f 5460
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