Theorem fnbrfvb 5642

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 2207	. . . 4 |- ( F ` B ) = ( F ` B )
2		funfvex 5616	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( F ` B ) e. _V )
3	2	funfni 5395	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. _V )
4		eqeq2 2217	. . . . . . . 8 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
5		breq2 4063	. . . . . . . 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 5399	. . . . . . 7 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
9		tz6.12c 5629	. . . . . . 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 2838	. . . . 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 4063	. . 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 5390	. . . 4 |- ( F Fn A -> Fun F )
17		funbrfv 5640	. . . 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 1373   E!weu 2055   e. wcel 2178   _Vcvv 2776   class class class wbr 4059   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  fnopfvb  5643  funbrfvb  5644  dffn5im  5647  fnsnfv  5661  fndmdif  5708  dffo4  5751  dff13  5860  isoini  5910  1stconst  6330  2ndconst  6331  znleval  14530  pw1nct  16142