Theorem funbrfv 5739

Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
funbrfv	|- ( Fun F -> ( A F B -> ( F ` A ) = B ) )

Proof of Theorem funbrfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 5394	. . . 4 |- ( Fun F -> Rel F )
2		brrelex2 4816	. . . 4 |- ( ( Rel F /\ A F B ) -> B e. _V )
3	1, 2	sylan 283	. . 3 |- ( ( Fun F /\ A F B ) -> B e. _V )
4		breq2 4134	. . . . . 6 |- ( y = B -> ( A F y <-> A F B ) )
5	4	anbi2d 468	. . . . 5 |- ( y = B -> ( ( Fun F /\ A F y ) <-> ( Fun F /\ A F B ) ) )
6		eqeq2 2248	. . . . 5 |- ( y = B -> ( ( F ` A ) = y <-> ( F ` A ) = B ) )
7	5, 6	imbi12d 234	. . . 4 |- ( y = B -> ( ( ( Fun F /\ A F y ) -> ( F ` A ) = y ) <-> ( ( Fun F /\ A F B ) -> ( F ` A ) = B ) ) )
8		funeu 5402	. . . . . 6 |- ( ( Fun F /\ A F y ) -> E! y A F y )
9		tz6.12-1 5722	. . . . . 6 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )
10	8, 9	sylan2 286	. . . . 5 |- ( ( A F y /\ ( Fun F /\ A F y ) ) -> ( F ` A ) = y )
11	10	anabss7 589	. . . 4 |- ( ( Fun F /\ A F y ) -> ( F ` A ) = y )
12	7, 11	vtoclg 2883	. . 3 |- ( B e. _V -> ( ( Fun F /\ A F B ) -> ( F ` A ) = B ) )
13	3, 12	mpcom 36	. 2 |- ( ( Fun F /\ A F B ) -> ( F ` A ) = B )
14	13	ex 115	1 |- ( Fun F -> ( A F B -> ( F ` A ) = B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E!weu 2086   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   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-fv 5385

This theorem is used by:  funopfv  5740  fnbrfvb  5741  fvelima  5754  fvi  5760  fmptco  5874  fliftfun  6002  fliftval  6006  tfrlem5  6585  sum0  12155  isumz  12156  fsumsersdc  12162  isumclim  12188  zprodap0  12348  dvaddxx  15804  dvmulxx  15805  dvcj  15810  dvrecap  15814  dvef  15828  pilem3  15884