Theorem erlecpbl 12811

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 1020	. . . 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 1020	. . . 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 1025	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
7		simp3l 1027	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
8	2, 4, 5, 6, 7	ercpbllemg 12809	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A .~ C ) )
9		simp2r 1026	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10		simp3r 1028	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11	2, 4, 5, 9, 10	ercpbllemg 12809	. . 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 1020	. 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 980   = wceq 1364   e. wcel 2160   class class class wbr 4018   |-> cmpt 4079   ` cfv 5235   Er wer 6557   [cec 6558

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fv 5243  df-er 6560  df-ec 6562

This theorem is referenced by: (None)