Theorem erth 6173

Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
erth.1	|- ( ph -> R Er X )
erth.2	|- ( ph -> A e. X )

Assertion

Ref	Expression
erth	|- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Proof of Theorem erth

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 107	. . . . . . 7 |- ( ( ph /\ A R B ) -> ph )
2		erth.1	. . . . . . . . 9 |- ( ph -> R Er X )
3	2	ersymb 6143	. . . . . . . 8 |- ( ph -> ( A R B <-> B R A ) )
4	3	biimpa 290	. . . . . . 7 |- ( ( ph /\ A R B ) -> B R A )
5	1, 4	jca 300	. . . . . 6 |- ( ( ph /\ A R B ) -> ( ph /\ B R A ) )
6	2	ertr 6144	. . . . . . 7 |- ( ph -> ( ( B R A /\ A R x ) -> B R x ) )
7	6	impl 372	. . . . . 6 |- ( ( ( ph /\ B R A ) /\ A R x ) -> B R x )
8	5, 7	sylan 277	. . . . 5 |- ( ( ( ph /\ A R B ) /\ A R x ) -> B R x )
9	2	ertr 6144	. . . . . 6 |- ( ph -> ( ( A R B /\ B R x ) -> A R x ) )
10	9	impl 372	. . . . 5 |- ( ( ( ph /\ A R B ) /\ B R x ) -> A R x )
11	8, 10	impbida 560	. . . 4 |- ( ( ph /\ A R B ) -> ( A R x <-> B R x ) )
12		vex 2604	. . . . 5 |- x e. _V
13		erth.2	. . . . . 6 |- ( ph -> A e. X )
14	13	adantr 270	. . . . 5 |- ( ( ph /\ A R B ) -> A e. X )
15		elecg 6167	. . . . 5 |- ( ( x e. _V /\ A e. X ) -> ( x e. [ A ] R <-> A R x ) )
16	12, 14, 15	sylancr 405	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> A R x ) )
17		errel 6138	. . . . . . 7 |- ( R Er X -> Rel R )
18	2, 17	syl 14	. . . . . 6 |- ( ph -> Rel R )
19		brrelex2 4401	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
20	18, 19	sylan 277	. . . . 5 |- ( ( ph /\ A R B ) -> B e. _V )
21		elecg 6167	. . . . 5 |- ( ( x e. _V /\ B e. _V ) -> ( x e. [ B ] R <-> B R x ) )
22	12, 20, 21	sylancr 405	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ B ] R <-> B R x ) )
23	11, 16, 22	3bitr4d 218	. . 3 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> x e. [ B ] R ) )
24	23	eqrdv 2079	. 2 |- ( ( ph /\ A R B ) -> [ A ] R = [ B ] R )
25	2	adantr 270	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> R Er X )
26	2, 13	erref 6149	. . . . . . 7 |- ( ph -> A R A )
27	26	adantr 270	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R A )
28	13	adantr 270	. . . . . . 7 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. X )
29		elecg 6167	. . . . . . 7 |- ( ( A e. X /\ A e. X ) -> ( A e. [ A ] R <-> A R A ) )
30	28, 28, 29	syl2anc 403	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ A ] R <-> A R A ) )
31	27, 30	mpbird 165	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ A ] R )
32		simpr 108	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> [ A ] R = [ B ] R )
33	31, 32	eleqtrd 2157	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ B ] R )
34	25, 32	ereldm 6172	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. X <-> B e. X ) )
35	28, 34	mpbid 145	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B e. X )
36		elecg 6167	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A e. [ B ] R <-> B R A ) )
37	28, 35, 36	syl2anc 403	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ B ] R <-> B R A ) )
38	33, 37	mpbid 145	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B R A )
39	25, 38	ersym 6141	. 2 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R B )
40	24, 39	impbida 560	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   Rel wrel 4368   Er wer 6126   [cec 6127

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-er 6129  df-ec 6131

This theorem is referenced by:  erth2  6174  erthi  6175  qliftfun  6211  eroveu  6220  th3qlem1  6231  enqeceq  6549  enq0eceq  6627  nnnq0lem1  6636  enreceq  6913  prsrlem1  6919