Theorem funbrfvb 5673

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 5347	. 2 |- ( Fun F <-> F Fn dom F )
2		fnbrfvb 5671	. 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 1395   e. wcel 2200   class class class wbr 4082   dom cdm 4718   Fun wfun 5311   Fn wfn 5312   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  funbrfv2b  5677  dfimafn  5681  funimass4  5683  dvidlemap  15359  dvidrelem  15360  dvidsslem  15361  pilem3  15451