Theorem fnbrfvb 5558

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 2177	. . . 4 |- ( F ` B ) = ( F ` B )
2		funfvex 5534	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. _V )
3	2	funfni 5318	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. _V )
4		eqeq2 2187	. . . . . . . 8 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
5		breq2 4009	. . . . . . . 8 |- ( x = ( F ` B ) -> ( B F x <-> B F ( F ` B ) ) )
6	4, 5	bibi12d 235	. . . . . . 7 |- ( x = ( F ` B ) -> ( ( ( F ` B ) = x <-> B F x ) <-> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) ) )
7	6	imbi2d 230	. . . . . 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 5322	. . . . . . 7 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
9		tz6.12c 5547	. . . . . . 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 2799	. . . . 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 148	. . 3 |- ( ( F Fn A /\ B e. A ) -> B F ( F ` B ) )
14		breq2 4009	. . 3 |- ( ( F ` B ) = C -> ( B F ( F ` B ) <-> B F C ) )
15	13, 14	syl5ibcom 155	. 2 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C -> B F C ) )
16		fnfun 5315	. . . 4 |- ( F Fn A -> Fun F )
17		funbrfv 5556	. . . 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 276	. 2 |- ( ( F Fn A /\ B e. A ) -> ( B F C -> ( F ` B ) = C ) )
20	15, 19	impbid 129	1 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E!weu 2026   e. wcel 2148   _Vcvv 2739   class class class wbr 4005   Fun wfun 5212   Fn wfn 5213   ` cfv 5218

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  fnopfvb  5559  funbrfvb  5560  dffn5im  5563  fnsnfv  5577  fndmdif  5623  dffo4  5666  dff13  5771  isoini  5821  1stconst  6224  2ndconst  6225  pw1nct  14837