Theorem fnbrafvb 41234

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6236. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
fnbrafvb	|- ( ( F Fn A /\ B e. A ) -> ( ( F''' B ) = C <-> B F C ) )

Proof of Theorem fnbrafvb

Step	Hyp	Ref	Expression
1		fndm 5990	. . . . . 6 |- ( F Fn A -> dom F = A )
2		eleq2 2690	. . . . . . . 8 |- ( A = dom F -> ( B e. A <-> B e. dom F ) )
3	2	eqcoms 2630	. . . . . . 7 |- ( dom F = A -> ( B e. A <-> B e. dom F ) )
4	3	biimpd 219	. . . . . 6 |- ( dom F = A -> ( B e. A -> B e. dom F ) )
5	1, 4	syl 17	. . . . 5 |- ( F Fn A -> ( B e. A -> B e. dom F ) )
6	5	imp 445	. . . 4 |- ( ( F Fn A /\ B e. A ) -> B e. dom F )
7		snssi 4339	. . . . . . 7 |- ( B e. A -> { B } C_ A )
8	7	adantl 482	. . . . . 6 |- ( ( F Fn A /\ B e. A ) -> { B } C_ A )
9		fnssresb 6003	. . . . . . 7 |- ( F Fn A -> ( ( F |` { B } ) Fn { B } <-> { B } C_ A ) )
10	9	adantr 481	. . . . . 6 |- ( ( F Fn A /\ B e. A ) -> ( ( F |` { B } ) Fn { B } <-> { B } C_ A ) )
11	8, 10	mpbird 247	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) Fn { B } )
12		fnfun 5988	. . . . 5 |- ( ( F |` { B } ) Fn { B } -> Fun ( F |` { B } ) )
13	11, 12	syl 17	. . . 4 |- ( ( F Fn A /\ B e. A ) -> Fun ( F |` { B } ) )
14		df-dfat 41196	. . . . 5 |- ( F defAt B <-> ( B e. dom F /\ Fun ( F |` { B } ) ) )
15		afvfundmfveq 41218	. . . . 5 |- ( F defAt B -> ( F''' B ) = ( F ` B ) )
16	14, 15	sylbir 225	. . . 4 |- ( ( B e. dom F /\ Fun ( F |` { B } ) ) -> ( F''' B ) = ( F ` B ) )
17	6, 13, 16	syl2anc 693	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( F''' B ) = ( F ` B ) )
18	17	eqeq1d 2624	. 2 |- ( ( F Fn A /\ B e. A ) -> ( ( F''' B ) = C <-> ( F ` B ) = C ) )
19		fnbrfvb 6236	. 2 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )
20	18, 19	bitrd 268	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   C_ wss 3574   {csn 4177   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   defAt wdfat 41193  '''cafv 41194

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-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  fnopafvb  41235  funbrafvb  41236  dfafn5a  41240