Theorem funbrfv2b 6240

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 5905	. . . 4 |- ( Fun F -> Rel F )
2		releldm 5358	. . . . 5 |- ( ( Rel F /\ A F B ) -> A e. dom F )
3	2	ex 450	. . . 4 |- ( Rel F -> ( A F B -> A e. dom F ) )
4	1, 3	syl 17	. . 3 |- ( Fun F -> ( A F B -> A e. dom F ) )
5	4	pm4.71rd 667	. 2 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ A F B ) ) )
6		funbrfvb 6238	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
7	6	pm5.32da 673	. 2 |- ( Fun F -> ( ( A e. dom F /\ ( F ` A ) = B ) <-> ( A e. dom F /\ A F B ) ) )
8	5, 7	bitr4d 271	1 |- ( Fun F -> ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  brtpos2  7358  mpt2curryd  7395  xpcomco  8050  fseqenlem2  8848  fpwwe2  9465  joinfval  17001  joinfval2  17002  meetfval  17015  meetfval2  17016  tayl0  24116  ofpreima  29465  funcnvmptOLD  29467  funcnvmpt  29468  curf  33387  uncf  33388  curunc  33391  fperdvper  40133