Theorem funopfv 6235

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

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A F B <-> <. A , B >. e. F )
2		funbrfv 6234	. 2 |- ( Fun F -> ( A F B -> ( F ` A ) = B ) )
3	1, 2	syl5bir 233	1 |- ( Fun F -> ( <. A , B >. e. F -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvopab3ig  6278  fvsn  6446  fveqf1o  6557  ovidig  6778  ovigg  6781  f1o2ndf1  7285  fundmen  8030  uzrdg0i  12758  uzrdgsuci  12759  strfvd  15904  strfv2d  15905  imasaddvallem  16189  imasvscafn  16197  basvtxvalOLD  25903  edgfiedgvalOLD  25904  adjeq  28794  bnj1379  30901  bnj97  30936  bnj553  30968  bnj966  31014  bnj1442  31117