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Theorem fmpt3d 26144
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 2454 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
31, 2fmptd 5979 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
4 fmpt3d.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
54feq1d 5657 . 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 1370    e. wcel 1758    |-> cmpt 4461   -->wf 5525
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537
This theorem is referenced by:  ofoprabco  26153  qqhf  26580  indf  26637  esumcocn  26694  ofcf  26710  mbfmcst  26838  dstrvprob  27018  dstfrvclim1  27024
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