Theorem funbrfv 5619

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 5289	. . . 4 |- ( Fun F -> Rel F )
2		brrelex2 4717	. . . 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 4049	. . . . . 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 2215	. . . . 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 5297	. . . . . 6 |- ( ( Fun F /\ A F y ) -> E! y A F y )
9		tz6.12-1 5605	. . . . . 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 2833	. . 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 1373   E!weu 2054   e. wcel 2176   _Vcvv 2772   class class class wbr 4045   Rel wrel 4681   Fun wfun 5266   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280

This theorem is referenced by:  funopfv  5620  fnbrfvb  5621  fvelima  5632  fvi  5638  fmptco  5748  fliftfun  5867  fliftval  5871  tfrlem5  6402  sum0  11732  isumz  11733  fsumsersdc  11739  isumclim  11765  zprodap0  11925  dvaddxx  15208  dvmulxx  15209  dvcj  15214  dvrecap  15218  dvef  15232  pilem3  15288