Theorem funbrfvb 5686

Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Assertion

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

Proof of Theorem funbrfvb

Step	Hyp	Ref	Expression
1		funfn 5356	. 2 |- ( Fun F <-> F Fn dom F )
2		fnbrfvb 5684	. 2 |- ( ( F Fn dom F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
3	1, 2	sylanb 284	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   dom cdm 4725   Fun wfun 5320   Fn wfn 5321   ` cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  funbrfv2b  5690  dfimafn  5694  funimass4  5696  dvidlemap  15414  dvidrelem  15415  dvidsslem  15416  pilem3  15506