Theorem funopfvb 6239

Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Assertion

Ref	Expression
funopfvb	|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )

Proof of Theorem funopfvb

Step	Hyp	Ref	Expression
1		funfn 5918	. 2 |- ( Fun F <-> F Fn dom F )
2		fnopfvb 6237	. 2 |- ( ( F Fn dom F /\ A e. dom F ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
3	1, 2	sylanb 489	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` 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-fn 5891  df-fv 5896

This theorem is referenced by:  dmfco  6272  funfvop  6329  f1eqcocnv  6556  usgredgop  26065  fgreu  29471