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

Theorem funopfv 5920
Description: The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
Assertion
Ref Expression
funopfv  |-  ( Fun 
F  ->  ( <. A ,  B >.  e.  F  ->  ( F `  A
)  =  B ) )

Proof of Theorem funopfv
StepHypRef Expression
1 df-br 4427 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
2 funbrfv 5919 . 2  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
31, 2syl5bir 221 1  |-  ( Fun 
F  ->  ( <. A ,  B >.  e.  F  ->  ( F `  A
)  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426   Fun wfun 5595   ` cfv 5601
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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
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-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-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  fvopab3ig  5961  fvsn  6112  fveqf1o  6215  ovidig  6428  ovigg  6431  f1o2ndf1  6915  fundmen  7650  uzrdg0i  12170  uzrdgsuci  12171  strfvd  15117  strfv2d  15118  imasaddvallem  15386  imasvscafn  15394  adjeq  27423  bnj1379  29430  bnj97  29465  bnj553  29497  bnj966  29543  bnj1442  29646
  Copyright terms: Public domain W3C validator