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Theorem fmpt3d 25924
Description: Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
fmpt3d.1  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
fmpt3d.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
Assertion
Ref Expression
fmpt3d  |-  ( ph  ->  F : A --> C )
Distinct variable groups:    x, A    x, C    ph, x
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fmpt3d
StepHypRef Expression
1 fmpt3d.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
2 eqid 2438 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
31, 2fmptd 5862 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
4 fmpt3d.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
54feq1d 5541 . 2  |-  ( ph  ->  ( F : A --> C 
<->  ( x  e.  A  |->  B ) : A --> C ) )
63, 5mpbird 232 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4345   -->wf 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421
This theorem is referenced by:  ofoprabco  25933  qqhf  26367  indf  26424  esumcocn  26481  ofcf  26497  mbfmcst  26626  dstrvprob  26806  dstfrvclim1  26812
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