Theorem funbrfv2b 5512

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 5186	. . . 4 |- ( Fun F -> Rel F )
2		releldm 4820	. . . . 5 |- ( ( Rel F /\ A F B ) -> A e. dom F )
3	2	ex 114	. . . 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 392	. 2 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ A F B ) ) )
6		funbrfvb 5510	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
7	6	pm5.32da 448	. 2 |- ( Fun F -> ( ( A e. dom F /\ ( F ` A ) = B ) <-> ( A e. dom F /\ A F B ) ) )
8	5, 7	bitr4d 190	1 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   class class class wbr 3965   dom cdm 4585   Rel wrel 4590   Fun wfun 5163   ` cfv 5169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fn 5172  df-fv 5177

This theorem is referenced by:  brtpos2  6195  xpcomco  6768