Theorem fnbrfvb 5604

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 2196	. . . 4 |- ( F ` B ) = ( F ` B )
2		funfvex 5578	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. _V )
3	2	funfni 5361	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. _V )
4		eqeq2 2206	. . . . . . . 8 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
5		breq2 4038	. . . . . . . 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 5365	. . . . . . 7 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
9		tz6.12c 5591	. . . . . . 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 2824	. . . . 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 4038	. . 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 5356	. . . 4 |- ( F Fn A -> Fun F )
17		funbrfv 5602	. . . 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 1364   E!weu 2045   e. wcel 2167   _Vcvv 2763   class class class wbr 4034   Fun wfun 5253   Fn wfn 5254   ` cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  fnopfvb  5605  funbrfvb  5606  dffn5im  5609  fnsnfv  5623  fndmdif  5670  dffo4  5713  dff13  5818  isoini  5868  1stconst  6288  2ndconst  6289  znleval  14285  pw1nct  15734