Theorem erth 7791

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 473	. . . . . . 7 |- ( ( ph /\ A R B ) -> ph )
2		erth.1	. . . . . . . . 9 |- ( ph -> R Er X )
3	2	ersymb 7756	. . . . . . . 8 |- ( ph -> ( A R B <-> B R A ) )
4	3	biimpa 501	. . . . . . 7 |- ( ( ph /\ A R B ) -> B R A )
5	1, 4	jca 554	. . . . . 6 |- ( ( ph /\ A R B ) -> ( ph /\ B R A ) )
6	2	ertr 7757	. . . . . . 7 |- ( ph -> ( ( B R A /\ A R x ) -> B R x ) )
7	6	impl 650	. . . . . 6 |- ( ( ( ph /\ B R A ) /\ A R x ) -> B R x )
8	5, 7	sylan 488	. . . . 5 |- ( ( ( ph /\ A R B ) /\ A R x ) -> B R x )
9	2	ertr 7757	. . . . . 6 |- ( ph -> ( ( A R B /\ B R x ) -> A R x ) )
10	9	impl 650	. . . . 5 |- ( ( ( ph /\ A R B ) /\ B R x ) -> A R x )
11	8, 10	impbida 877	. . . 4 |- ( ( ph /\ A R B ) -> ( A R x <-> B R x ) )
12		vex 3203	. . . . 5 |- x e. _V
13		erth.2	. . . . . 6 |- ( ph -> A e. X )
14	13	adantr 481	. . . . 5 |- ( ( ph /\ A R B ) -> A e. X )
15		elecg 7785	. . . . 5 |- ( ( x e. _V /\ A e. X ) -> ( x e. [ A ] R <-> A R x ) )
16	12, 14, 15	sylancr 695	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> A R x ) )
17		errel 7751	. . . . . . 7 |- ( R Er X -> Rel R )
18	2, 17	syl 17	. . . . . 6 |- ( ph -> Rel R )
19		brrelex2 5157	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
20	18, 19	sylan 488	. . . . 5 |- ( ( ph /\ A R B ) -> B e. _V )
21		elecg 7785	. . . . 5 |- ( ( x e. _V /\ B e. _V ) -> ( x e. [ B ] R <-> B R x ) )
22	12, 20, 21	sylancr 695	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ B ] R <-> B R x ) )
23	11, 16, 22	3bitr4d 300	. . 3 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> x e. [ B ] R ) )
24	23	eqrdv 2620	. 2 |- ( ( ph /\ A R B ) -> [ A ] R = [ B ] R )
25	2	adantr 481	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> R Er X )
26	2, 13	erref 7762	. . . . . . 7 |- ( ph -> A R A )
27	26	adantr 481	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R A )
28	13	adantr 481	. . . . . . 7 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. X )
29		elecg 7785	. . . . . . 7 |- ( ( A e. X /\ A e. X ) -> ( A e. [ A ] R <-> A R A ) )
30	28, 28, 29	syl2anc 693	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ A ] R <-> A R A ) )
31	27, 30	mpbird 247	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ A ] R )
32		simpr 477	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> [ A ] R = [ B ] R )
33	31, 32	eleqtrd 2703	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ B ] R )
34	25, 32	ereldm 7790	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. X <-> B e. X ) )
35	28, 34	mpbid 222	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B e. X )
36		elecg 7785	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A e. [ B ] R <-> B R A ) )
37	28, 35, 36	syl2anc 693	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ B ] R <-> B R A ) )
38	33, 37	mpbid 222	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B R A )
39	25, 38	ersym 7754	. 2 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R B )
40	24, 39	impbida 877	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Rel wrel 5119   Er wer 7739   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744

This theorem is referenced by:  erth2  7792  erthi  7793  qliftfun  7832  eroveu  7842  eceqoveq  7853  enreceq  9887  prsrlem1  9893  ercpbllem  16208  orbsta  17746  sylow2blem3  18037  frgpnabllem2  18277  zndvds  19898  qustgpopn  21923  qustgphaus  21926  pi1xfrf  22853  pi1cof  22859  pstmxmet  29940  sconnpi1  31221  topfneec2  32351