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

Theorem fpm 7512
Description: A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
elmap.1  |-  A  e. 
_V
elmap.2  |-  B  e. 
_V
Assertion
Ref Expression
fpm  |-  ( F : A --> B  ->  F  e.  ( B  ^pm  A ) )

Proof of Theorem fpm
StepHypRef Expression
1 elmap.1 . 2  |-  A  e. 
_V
2 elmap.2 . 2  |-  B  e. 
_V
3 fpmg 7505 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  F : A --> B )  ->  F  e.  ( B  ^pm  A ) )
41, 2, 3mp3an12 1350 1  |-  ( F : A --> B  ->  F  e.  ( B  ^pm  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087   -->wf 5597  (class class class)co 6305    ^pm cpm 7481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-pm 7483
This theorem is referenced by:  plycpn  23110
  Copyright terms: Public domain W3C validator