Theorem funbrfv2b 5721

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 5369	. . . 4 |- ( Fun F -> Rel F )
2		releldm 4992	. . . . 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 394	. 2 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ A F B ) ) )
6		funbrfvb 5717	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
7	6	pm5.32da 452	. 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109   dom cdm 4749   Rel wrel 4754   Fun wfun 5346   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  brtpos2  6482  xpcomco  7077