Theorem funbrfv 5212

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 4919	. . . 4 |- ( Fun F -> Rel F )
2		brrelex2 4383	. . . 4 |- ( ( Rel F /\ A F B ) -> B e. _V )
3	1, 2	sylan 267	. . 3 |- ( ( Fun F /\ A F B ) -> B e. _V )
4		breq2 3768	. . . . . 6 |- ( y = B -> ( A F y <-> A F B ) )
5	4	anbi2d 437	. . . . 5 |- ( y = B -> ( ( Fun F /\ A F y ) <-> ( Fun F /\ A F B ) ) )
6		eqeq2 2049	. . . . 5 |- ( y = B -> ( ( F ` A ) = y <-> ( F ` A ) = B ) )
7	5, 6	imbi12d 223	. . . 4 |- ( y = B -> ( ( ( Fun F /\ A F y ) -> ( F ` A ) = y ) <-> ( ( Fun F /\ A F B ) -> ( F ` A ) = B ) ) )
8		funeu 4926	. . . . . 6 |- ( ( Fun F /\ A F y ) -> E! y A F y )
9		tz6.12-1 5200	. . . . . 6 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )
10	8, 9	sylan2 270	. . . . 5 |- ( ( A F y /\ ( Fun F /\ A F y ) ) -> ( F ` A ) = y )
11	10	anabss7 517	. . . 4 |- ( ( Fun F /\ A F y ) -> ( F ` A ) = y )
12	7, 11	vtoclg 2613	. . 3 |- ( B e. _V -> ( ( Fun F /\ A F B ) -> ( F ` A ) = B ) )
13	3, 12	mpcom 32	. 2 |- ( ( Fun F /\ A F B ) -> ( F ` A ) = B )
14	13	ex 108	1 |- ( Fun F -> ( A F B -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   E!weu 1900   _Vcvv 2557   class class class wbr 3764   Rel wrel 4350   Fun wfun 4896   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by:  funopfv  5213  fnbrfvb  5214  fvelima  5225  fvi  5230  fmptco  5330  fliftfun  5436  fliftval  5440  tfrlem5  5930