Theorem fcoinvbr 29419

Description: Binary relation for the equivalence relation from fcoinver 29418. (Contributed by Thierry Arnoux, 3-Jan-2020.)

Hypothesis

Ref	Expression
fcoinvbr.e	|- .~ = ( `' F o. F )

Assertion

Ref	Expression
fcoinvbr	|- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( X .~ Y <-> ( F ` X ) = ( F ` Y ) ) )

Proof of Theorem fcoinvbr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fcoinvbr.e	. . . . 5 |- .~ = ( `' F o. F )
2	1	breqi 4659	. . . 4 |- ( X .~ Y <-> X ( `' F o. F ) Y )
3		brcog 5288	. . . 4 |- ( ( X e. A /\ Y e. A ) -> ( X ( `' F o. F ) Y <-> E. z ( X F z /\ z `' F Y ) ) )
4	2, 3	syl5bb 272	. . 3 |- ( ( X e. A /\ Y e. A ) -> ( X .~ Y <-> E. z ( X F z /\ z `' F Y ) ) )
5	4	3adant1 1079	. 2 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( X .~ Y <-> E. z ( X F z /\ z `' F Y ) ) )
6		fvex 6201	. . . . 5 |- ( F ` X ) e. _V
7	6	eqvinc 3330	. . . 4 |- ( ( F ` X ) = ( F ` Y ) <-> E. z ( z = ( F ` X ) /\ z = ( F ` Y ) ) )
8		eqcom 2629	. . . . . 6 |- ( z = ( F ` X ) <-> ( F ` X ) = z )
9		eqcom 2629	. . . . . 6 |- ( z = ( F ` Y ) <-> ( F ` Y ) = z )
10	8, 9	anbi12i 733	. . . . 5 |- ( ( z = ( F ` X ) /\ z = ( F ` Y ) ) <-> ( ( F ` X ) = z /\ ( F ` Y ) = z ) )
11	10	exbii 1774	. . . 4 |- ( E. z ( z = ( F ` X ) /\ z = ( F ` Y ) ) <-> E. z ( ( F ` X ) = z /\ ( F ` Y ) = z ) )
12	7, 11	bitri 264	. . 3 |- ( ( F ` X ) = ( F ` Y ) <-> E. z ( ( F ` X ) = z /\ ( F ` Y ) = z ) )
13		fnbrfvb 6236	. . . . . . 7 |- ( ( F Fn A /\ X e. A ) -> ( ( F ` X ) = z <-> X F z ) )
14	13	3adant3 1081	. . . . . 6 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( F ` X ) = z <-> X F z ) )
15		fnbrfvb 6236	. . . . . . 7 |- ( ( F Fn A /\ Y e. A ) -> ( ( F ` Y ) = z <-> Y F z ) )
16	15	3adant2 1080	. . . . . 6 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( F ` Y ) = z <-> Y F z ) )
17	14, 16	anbi12d 747	. . . . 5 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( ( F ` X ) = z /\ ( F ` Y ) = z ) <-> ( X F z /\ Y F z ) ) )
18		vex 3203	. . . . . . . 8 |- z e. _V
19		brcnvg 5303	. . . . . . . 8 |- ( ( z e. _V /\ Y e. A ) -> ( z `' F Y <-> Y F z ) )
20	18, 19	mpan 706	. . . . . . 7 |- ( Y e. A -> ( z `' F Y <-> Y F z ) )
21	20	3ad2ant3 1084	. . . . . 6 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( z `' F Y <-> Y F z ) )
22	21	anbi2d 740	. . . . 5 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( X F z /\ z `' F Y ) <-> ( X F z /\ Y F z ) ) )
23	17, 22	bitr4d 271	. . . 4 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( ( F ` X ) = z /\ ( F ` Y ) = z ) <-> ( X F z /\ z `' F Y ) ) )
24	23	exbidv 1850	. . 3 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( E. z ( ( F ` X ) = z /\ ( F ` Y ) = z ) <-> E. z ( X F z /\ z `' F Y ) ) )
25	12, 24	syl5bb 272	. 2 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( ( F ` X ) = ( F ` Y ) <-> E. z ( X F z /\ z `' F Y ) ) )
26	5, 25	bitr4d 271	1 |- ( ( F Fn A /\ X e. A /\ Y e. A ) -> ( X .~ Y <-> ( F ` X ) = ( F ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   o. ccom 5118   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:  qtophaus  29903