Theorem fnbrfvb 5598

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 2193	. . . 4 |- ( F ` B ) = ( F ` B )
2		funfvex 5572	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. _V )
3	2	funfni 5355	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. _V )
4		eqeq2 2203	. . . . . . . 8 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
5		breq2 4034	. . . . . . . 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 5359	. . . . . . 7 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
9		tz6.12c 5585	. . . . . . 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 2821	. . . . 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 4034	. . 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 5352	. . . 4 |- ( F Fn A -> Fun F )
17		funbrfv 5596	. . . 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 2042   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   Fun wfun 5249   Fn wfn 5250   ` cfv 5255

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  fnopfvb  5599  funbrfvb  5600  dffn5im  5603  fnsnfv  5617  fndmdif  5664  dffo4  5707  dff13  5812  isoini  5862  1stconst  6276  2ndconst  6277  znleval  14152  pw1nct  15563