Theorem funbrfv 5554

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 5233	. . . 4 |- ( Fun F -> Rel F )
2		brrelex2 4667	. . . 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 4007	. . . . . 6 |- ( y = B -> ( A F y <-> A F B ) )
5	4	anbi2d 464	. . . . 5 |- ( y = B -> ( ( Fun F /\ A F y ) <-> ( Fun F /\ A F B ) ) )
6		eqeq2 2187	. . . . 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 5241	. . . . . 6 |- ( ( Fun F /\ A F y ) -> E! y A F y )
9		tz6.12-1 5542	. . . . . 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 583	. . . 4 |- ( ( Fun F /\ A F y ) -> ( F ` A ) = y )
12	7, 11	vtoclg 2797	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E!weu 2026   e. wcel 2148   _Vcvv 2737   class class class wbr 4003   Rel wrel 4631   Fun wfun 5210   ` cfv 5216

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224

This theorem is referenced by:  funopfv  5555  fnbrfvb  5556  fvelima  5567  fvi  5573  fmptco  5682  fliftfun  5796  fliftval  5800  tfrlem5  6314  sum0  11391  isumz  11392  fsumsersdc  11398  isumclim  11424  zprodap0  11584  dvaddxx  14060  dvmulxx  14061  dvcj  14066  dvrecap  14070  dvef  14081  pilem3  14097