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Theorem feq2 5376
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq2  |-  ( A  =  B  ->  ( F : A --> C  <->  F : B
--> C ) )

Proof of Theorem feq2
StepHypRef Expression
1 fneq2 5334 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 685 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  C_  C
)  <->  ( F  Fn  B  /\  ran  F  C_  C ) ) )
3 df-f 5259 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
4 df-f 5259 . 2  |-  ( F : B --> C  <->  ( F  Fn  B  /\  ran  F  C_  C ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : A --> C  <->  F : B
--> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152   ran crn 4690    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  feq23  5378  feq2d  5380  feq2i  5384  f00  5426  f1eq2  5433  fressnfv  5707  fconstfv  5734  mapvalg  6782  map0g  6807  ac6sfi  7101  cofsmo  7895  axcc4dom  8067  ac6sg  8115  isghm  14683  pjdm2  16611  cmpcovf  17118  ulmval  19759  measval  23529  isrnmeas  23531  poseq  24253  soseq  24254  elno2  24308  noreson  24314  noxpsgn  24319  nodenselem6  24340  intcont  25543  istopx  25547  algi  25727  stoweidlem62  27811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2276  df-fn 5258  df-f 5259
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