Theorem erthdmg 15730

Description: Equality of equivalence classes.

Assertion

Ref	Expression
erthdmg	|- ((B e. C /\ Er R /\ A e. dom R) -> ([A]R = [B]R <-> ARB))

Proof of Theorem erthdmg

Step	Hyp	Ref	Expression
1		eceq2 5336	. . . . . . 7 |- (B = if(B e. _V, B, (/)) -> [B]R = [if(B e. _V, B, (/))]R)
2	1	eqeq2d 1895	. . . . . 6 |- (B = if(B e. _V, B, (/)) -> ([A]R = [B]R <-> [A]R = [if(B e. _V, B, (/))]R))
3		breq2 3342	. . . . . 6 |- (B = if(B e. _V, B, (/)) -> (ARB <-> ARif(B e. _V, B, (/))))
4	2, 3	bibi12d 691	. . . . 5 |- (B = if(B e. _V, B, (/)) -> (([A]R = [B]R <-> ARB) <-> ([A]R = [if(B e. _V, B, (/))]R <-> ARif(B e. _V, B, (/)))))
5	4	imbi2d 674	. . . 4 |- (B = if(B e. _V, B, (/)) -> ((A e. dom R -> ([A]R = [B]R <-> ARB)) <-> (A e. dom R -> ([A]R = [if(B e. _V, B, (/))]R <-> ARif(B e. _V, B, (/))))))
6		dmeq 4157	. . . . . 6 |- (R = if(Er R, R, _I ) -> dom R = dom if(Er R, R, _I ))
7	6	eleq2d 1964	. . . . 5 |- (R = if(Er R, R, _I ) -> (A e. dom R <-> A e. dom if(Er R, R, _I )))
8		eceq1 5335	. . . . . . 7 |- (R = if(Er R, R, _I ) -> [A]R = [A]if(Er R, R, _I ))
9		eceq1 5335	. . . . . . 7 |- (R = if(Er R, R, _I ) -> [if(B e. _V, B, (/))]R = [if(B e. _V, B, (/))]if(Er R, R, _I ))
10	8, 9	eqeq12d 1899	. . . . . 6 |- (R = if(Er R, R, _I ) -> ([A]R = [if(B e. _V, B, (/))]R <-> [A]if(Er R, R, _I ) = [if(B e. _V, B, (/))]if(Er R, R, _I )))
11		breq 3340	. . . . . 6 |- (R = if(Er R, R, _I ) -> (ARif(B e. _V, B, (/)) <-> Aif(Er R, R, _I )if(B e. _V, B, (/))))
12	10, 11	bibi12d 691	. . . . 5 |- (R = if(Er R, R, _I ) -> (([A]R = [if(B e. _V, B, (/))]R <-> ARif(B e. _V, B, (/))) <-> ([A]if(Er R, R, _I ) = [if(B e. _V, B, (/))]if(Er R, R, _I ) <-> Aif(Er R, R, _I )if(B e. _V, B, (/)))))
13	7, 12	imbi12d 688	. . . 4 |- (R = if(Er R, R, _I ) -> ((A e. dom R -> ([A]R = [if(B e. _V, B, (/))]R <-> ARif(B e. _V, B, (/)))) <-> (A e. dom if(Er R, R, _I ) -> ([A]if(Er R, R, _I ) = [if(B e. _V, B, (/))]if(Er R, R, _I ) <-> Aif(Er R, R, _I )if(B e. _V, B, (/))))))
14		0ex 3446	. . . . . 6 |- (/) e. _V
15	14	elimel 3025	. . . . 5 |- if(B e. _V, B, (/)) e. _V
16		ereq 5324	. . . . . 6 |- (R = if(Er R, R, _I ) -> (Er R <-> Er if(Er R, R, _I )))
17		ereq 5324	. . . . . 6 |- ( _I = if(Er R, R, _I ) -> (Er _I <-> Er if(Er R, R, _I )))
18		ider 5326	. . . . . 6 |- Er _I
19	16, 17, 18	elimhyp 3021	. . . . 5 |- Er if(Er R, R, _I )
20	15, 19	erthdm 5341	. . . 4 |- (A e. dom if(Er R, R, _I ) -> ([A]if(Er R, R, _I ) = [if(B e. _V, B, (/))]if(Er R, R, _I ) <-> Aif(Er R, R, _I )if(B e. _V, B, (/))))
21	5, 13, 20	dedth2h 3015	. . 3 |- ((B e. _V /\ Er R) -> (A e. dom R -> ([A]R = [B]R <-> ARB)))
22		elisset 2299	. . 3 |- (B e. C -> B e. _V)
23	21, 22	sylan 497	. 2 |- ((B e. C /\ Er R) -> (A e. dom R -> ([A]R = [B]R <-> ARB)))
24	23	3impia 1064	1 |- ((B e. C /\ Er R /\ A e. dom R) -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982   class class class wbr 3338   _I cid 3582  dom cdm 3986  Er wer 5315  [cec 5316

This theorem is referenced by:  eropreu 15733  pi1gp 16095

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-er 5318  df-ec 5320

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