Theorem funbrfv2b 5747

Description: Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Assertion

Ref	Expression
funbrfv2b	|- ( Fun F -> ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) )

Proof of Theorem funbrfv2b

Step	Hyp	Ref	Expression
1		funrel 5394	. . . 4 |- ( Fun F -> Rel F )
2		releldm 5017	. . . . 5 |- ( ( Rel F /\ A F B ) -> A e. dom F )
3	2	ex 115	. . . 4 |- ( Rel F -> ( A F B -> A e. dom F ) )
4	1, 3	syl 14	. . 3 |- ( Fun F -> ( A F B -> A e. dom F ) )
5	4	pm4.71rd 398	. 2 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ A F B ) ) )
6		funbrfvb 5743	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
7	6	pm5.32da 456	. 2 |- ( Fun F -> ( ( A e. dom F /\ ( F ` A ) = B ) <-> ( A e. dom F /\ A F B ) ) )
8	5, 7	bitr4d 191	1 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   dom cdm 4774   Rel wrel 4779   Fun wfun 5371   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  brtpos2  6522  xpcomco  7124