Theorem erth 6150

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 102	. . . . . . 7 |- ( ( ph /\ A R B ) -> ph )
2		erth.1	. . . . . . . . 9 |- ( ph -> R Er X )
3	2	ersymb 6120	. . . . . . . 8 |- ( ph -> ( A R B <-> B R A ) )
4	3	biimpa 280	. . . . . . 7 |- ( ( ph /\ A R B ) -> B R A )
5	1, 4	jca 290	. . . . . 6 |- ( ( ph /\ A R B ) -> ( ph /\ B R A ) )
6	2	ertr 6121	. . . . . . 7 |- ( ph -> ( ( B R A /\ A R x ) -> B R x ) )
7	6	impl 362	. . . . . 6 |- ( ( ( ph /\ B R A ) /\ A R x ) -> B R x )
8	5, 7	sylan 267	. . . . 5 |- ( ( ( ph /\ A R B ) /\ A R x ) -> B R x )
9	2	ertr 6121	. . . . . 6 |- ( ph -> ( ( A R B /\ B R x ) -> A R x ) )
10	9	impl 362	. . . . 5 |- ( ( ( ph /\ A R B ) /\ B R x ) -> A R x )
11	8, 10	impbida 528	. . . 4 |- ( ( ph /\ A R B ) -> ( A R x <-> B R x ) )
12		vex 2560	. . . . 5 |- x e. _V
13		erth.2	. . . . . 6 |- ( ph -> A e. X )
14	13	adantr 261	. . . . 5 |- ( ( ph /\ A R B ) -> A e. X )
15		elecg 6144	. . . . 5 |- ( ( x e. _V /\ A e. X ) -> ( x e. [ A ] R <-> A R x ) )
16	12, 14, 15	sylancr 393	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> A R x ) )
17		errel 6115	. . . . . . 7 |- ( R Er X -> Rel R )
18	2, 17	syl 14	. . . . . 6 |- ( ph -> Rel R )
19		brrelex2 4383	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
20	18, 19	sylan 267	. . . . 5 |- ( ( ph /\ A R B ) -> B e. _V )
21		elecg 6144	. . . . 5 |- ( ( x e. _V /\ B e. _V ) -> ( x e. [ B ] R <-> B R x ) )
22	12, 20, 21	sylancr 393	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ B ] R <-> B R x ) )
23	11, 16, 22	3bitr4d 209	. . 3 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> x e. [ B ] R ) )
24	23	eqrdv 2038	. 2 |- ( ( ph /\ A R B ) -> [ A ] R = [ B ] R )
25	2	adantr 261	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> R Er X )
26	2, 13	erref 6126	. . . . . . 7 |- ( ph -> A R A )
27	26	adantr 261	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R A )
28	13	adantr 261	. . . . . . 7 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. X )
29		elecg 6144	. . . . . . 7 |- ( ( A e. X /\ A e. X ) -> ( A e. [ A ] R <-> A R A ) )
30	28, 28, 29	syl2anc 391	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ A ] R <-> A R A ) )
31	27, 30	mpbird 156	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ A ] R )
32		simpr 103	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> [ A ] R = [ B ] R )
33	31, 32	eleqtrd 2116	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ B ] R )
34	25, 32	ereldm 6149	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. X <-> B e. X ) )
35	28, 34	mpbid 135	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B e. X )
36		elecg 6144	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A e. [ B ] R <-> B R A ) )
37	28, 35, 36	syl2anc 391	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ B ] R <-> B R A ) )
38	33, 37	mpbid 135	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B R A )
39	25, 38	ersym 6118	. 2 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R B )
40	24, 39	impbida 528	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   Rel wrel 4350   Er wer 6103   [cec 6104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-er 6106  df-ec 6108

This theorem is referenced by:  erth2  6151  erthi  6152  qliftfun  6188  eroveu  6197  th3qlem1  6208  enqeceq  6457  enq0eceq  6535  nnnq0lem1  6544  enreceq  6821  prsrlem1  6827