Theorem fnbrfvb 5414

Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fnbrfvb	|- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )

Proof of Theorem fnbrfvb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2113	. . . 4 |- ( F ` B ) = ( F ` B )
2		funfvex 5390	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. _V )
3	2	funfni 5179	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. _V )
4		eqeq2 2122	. . . . . . . 8 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
5		breq2 3897	. . . . . . . 8 |- ( x = ( F ` B ) -> ( B F x <-> B F ( F ` B ) ) )
6	4, 5	bibi12d 234	. . . . . . 7 |- ( x = ( F ` B ) -> ( ( ( F ` B ) = x <-> B F x ) <-> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) ) )
7	6	imbi2d 229	. . . . . 6 |- ( x = ( F ` B ) -> ( ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = x <-> B F x ) ) <-> ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) ) ) )
8		fneu 5183	. . . . . . 7 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
9		tz6.12c 5403	. . . . . . 7 |- ( E! x B F x -> ( ( F ` B ) = x <-> B F x ) )
10	8, 9	syl 14	. . . . . 6 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = x <-> B F x ) )
11	7, 10	vtoclg 2715	. . . . 5 |- ( ( F ` B ) e. _V -> ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) ) )
12	3, 11	mpcom 36	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) )
13	1, 12	mpbii 147	. . 3 |- ( ( F Fn A /\ B e. A ) -> B F ( F ` B ) )
14		breq2 3897	. . 3 |- ( ( F ` B ) = C -> ( B F ( F ` B ) <-> B F C ) )
15	13, 14	syl5ibcom 154	. 2 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C -> B F C ) )
16		fnfun 5176	. . . 4 |- ( F Fn A -> Fun F )
17		funbrfv 5412	. . . 4 |- ( Fun F -> ( B F C -> ( F ` B ) = C ) )
18	16, 17	syl 14	. . 3 |- ( F Fn A -> ( B F C -> ( F ` B ) = C ) )
19	18	adantr 272	. 2 |- ( ( F Fn A /\ B e. A ) -> ( B F C -> ( F ` B ) = C ) )
20	15, 19	impbid 128	1 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   E!weu 1973   _Vcvv 2655   class class class wbr 3893   Fun wfun 5073   Fn wfn 5074   ` cfv 5079

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087

This theorem is referenced by:  fnopfvb  5415  funbrfvb  5416  dffn5im  5419  fnsnfv  5432  fndmdif  5477  dffo4  5520  dff13  5621  isoini  5671  1stconst  6069  2ndconst  6070