Theorem erlecpbl 13495

Description: Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	|- ( ph -> .~ Er V )
ercpbl.v	|- ( ph -> V e. W )
ercpbl.f	|- F = ( x e. V |-> [ x ] .~ )
erlecpbl.e	|- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A N B <-> C N D ) ) )

Assertion

Ref	Expression
erlecpbl	|- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )

Distinct variable groups:   x, .~   x, A   x, B   x, C   x, D   x, V   ph, x

Allowed substitution hints:   F( x)   N( x)   W( x)

Proof of Theorem erlecpbl

Step	Hyp	Ref	Expression
1		ercpbl.r	. . . . 5 |- ( ph -> .~ Er V )
2	1	3ad2ant1 1045	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> .~ Er V )
3		ercpbl.v	. . . . 5 |- ( ph -> V e. W )
4	3	3ad2ant1 1045	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> V e. W )
5		ercpbl.f	. . . 4 |- F = ( x e. V |-> [ x ] .~ )
6		simp2l 1050	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
7		simp3l 1052	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
8	2, 4, 5, 6, 7	ercpbllemg 13493	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A .~ C ) )
9		simp2r 1051	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10		simp3r 1053	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11	2, 4, 5, 9, 10	ercpbllemg 13493	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B .~ D ) )
12	8, 11	anbi12d 473	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) <-> ( A .~ C /\ B .~ D ) ) )
13		erlecpbl.e	. . 3 |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A N B <-> C N D ) ) )
14	13	3ad2ant1 1045	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( A .~ C /\ B .~ D ) -> ( A N B <-> C N D ) ) )
15	12, 14	sylbid 150	1 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333   Er wer 6742   [cec 6743

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fv 5341  df-er 6745  df-ec 6747

This theorem is referenced by: (None)