Theorem funbrfv 5742

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 5397	. . . 4 |- ( Fun F -> Rel F )
2		brrelex2 4818	. . . 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 4135	. . . . . 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 5405	. . . . . 6 |- ( ( Fun F /\ A F y ) -> E! y A F y )
9		tz6.12-1 5725	. . . . . 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 4131   Rel wrel 4781   Fun wfun 5374   ` cfv 5380

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 4251  ax-pow 4313  ax-pr 4348

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 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388

This theorem is used by:  funopfv  5743  fnbrfvb  5744  fvelima  5757  fvi  5763  fmptco  5877  fliftfun  6005  fliftval  6009  tfrlem5  6589  sum0  12180  isumz  12181  fsumsersdc  12187  isumclim  12213  zprodap0  12373  dvaddxx  15929  dvmulxx  15930  dvcj  15935  dvrecap  15939  dvef  15953  pilem3  16010