MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpt3d Structured version   Unicode version

Theorem fmpt3d 6033
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 2402 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
31, 2fmptd 6032 . 2  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
4 fmpt3d.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
54feq1d 5699 . 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 367    = wceq 1405    e. wcel 1842    |-> cmpt 4452   -->wf 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576
This theorem is referenced by:  ofoprabco  27935  sgnsf  28157  qqhf  28405  indf  28449  esumcocn  28513  ofcf  28536  mbfmcst  28693  dstrvprob  28902  dstfrvclim1  28908  signstf  29015  binomcxplemnotnn0  36089
  Copyright terms: Public domain W3C validator