Theorem fnbrfvb 6236

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 2622	. . . 4 |- ( F ` B ) = ( F ` B )
2		fvex 6201	. . . . 5 |- ( F ` B ) e. _V
3		eqeq2 2633	. . . . . . 7 |- ( x = ( F ` B ) -> ( ( F ` B ) = x <-> ( F ` B ) = ( F ` B ) ) )
4		breq2 4657	. . . . . . 7 |- ( x = ( F ` B ) -> ( B F x <-> B F ( F ` B ) ) )
5	3, 4	bibi12d 335	. . . . . 6 |- ( x = ( F ` B ) -> ( ( ( F ` B ) = x <-> B F x ) <-> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) ) )
6	5	imbi2d 330	. . . . 5 |- ( 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 ) ) ) ) )
7		fneu 5995	. . . . . 6 |- ( ( F Fn A /\ B e. A ) -> E! x B F x )
8		tz6.12c 6213	. . . . . 6 |- ( E! x B F x -> ( ( F ` B ) = x <-> B F x ) )
9	7, 8	syl 17	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = x <-> B F x ) )
10	2, 6, 9	vtocl 3259	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = ( F ` B ) <-> B F ( F ` B ) ) )
11	1, 10	mpbii 223	. . 3 |- ( ( F Fn A /\ B e. A ) -> B F ( F ` B ) )
12		breq2 4657	. . 3 |- ( ( F ` B ) = C -> ( B F ( F ` B ) <-> B F C ) )
13	11, 12	syl5ibcom 235	. 2 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C -> B F C ) )
14		fnfun 5988	. . . 4 |- ( F Fn A -> Fun F )
15		funbrfv 6234	. . . 4 |- ( Fun F -> ( B F C -> ( F ` B ) = C ) )
16	14, 15	syl 17	. . 3 |- ( F Fn A -> ( B F C -> ( F ` B ) = C ) )
17	16	adantr 481	. 2 |- ( ( F Fn A /\ B e. A ) -> ( B F C -> ( F ` B ) = C ) )
18	13, 17	impbid 202	1 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   class class class wbr 4653   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fnopfvb  6237  funbrfvb  6238  dffn5  6241  feqmptdf  6251  fnsnfv  6258  fndmdif  6321  dffo4  6375  dff13  6512  isomin  6587  isoini  6588  1stconst  7265  2ndconst  7266  fsplit  7282  seqomlem3  7547  seqomlem4  7548  nqerrel  9754  imasleval  16201  znleval  19903  axcontlem5  25848  elnlfn  28787  adjbd1o  28944  fcoinvbr  29419  elintfv  31662  br1steq  31670  br2ndeq  31671  br1steqg  31672  br2ndeqg  31673  fv1stcnv  31680  fv2ndcnv  31681  trpredpred  31728  scutun12  31917  madeval2  31936  fvbigcup  32009  fvsingle  32027  imageval  32037  brfullfun  32055  bj-mptval  33070  unccur  33392  poimirlem2  33411  poimirlem23  33432  pw2f1ocnv  37604  brcoffn  38328  funressnfv  41208  fnbrafvb  41234