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Theorem fmpt3d 6047
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 2451 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
31, 2fmptd 6046 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
4 fmpt3d.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
54feq1d 5714 . 2  |-  ( ph  ->  ( F : A --> C 
<->  ( x  e.  A  |->  B ) : A --> C ) )
63, 5mpbird 236 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    |-> cmpt 4461   -->wf 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590
This theorem is referenced by:  nmof  21724  ofoprabco  28267  sgnsf  28492  qqhf  28790  indf  28837  esumcocn  28901  ofcf  28924  mbfmcst  29081  dstrvprob  29304  dstfrvclim1  29310  signstf  29455  binomcxplemnotnn0  36705  hoicvrrex  38378
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