Theorem erth 6557

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 108	. . . . . . 7 |- ( ( ph /\ A R B ) -> ph )
2		erth.1	. . . . . . . . 9 |- ( ph -> R Er X )
3	2	ersymb 6527	. . . . . . . 8 |- ( ph -> ( A R B <-> B R A ) )
4	3	biimpa 294	. . . . . . 7 |- ( ( ph /\ A R B ) -> B R A )
5	1, 4	jca 304	. . . . . 6 |- ( ( ph /\ A R B ) -> ( ph /\ B R A ) )
6	2	ertr 6528	. . . . . . 7 |- ( ph -> ( ( B R A /\ A R x ) -> B R x ) )
7	6	impl 378	. . . . . 6 |- ( ( ( ph /\ B R A ) /\ A R x ) -> B R x )
8	5, 7	sylan 281	. . . . 5 |- ( ( ( ph /\ A R B ) /\ A R x ) -> B R x )
9	2	ertr 6528	. . . . . 6 |- ( ph -> ( ( A R B /\ B R x ) -> A R x ) )
10	9	impl 378	. . . . 5 |- ( ( ( ph /\ A R B ) /\ B R x ) -> A R x )
11	8, 10	impbida 591	. . . 4 |- ( ( ph /\ A R B ) -> ( A R x <-> B R x ) )
12		vex 2733	. . . . 5 |- x e. _V
13		erth.2	. . . . . 6 |- ( ph -> A e. X )
14	13	adantr 274	. . . . 5 |- ( ( ph /\ A R B ) -> A e. X )
15		elecg 6551	. . . . 5 |- ( ( x e. _V /\ A e. X ) -> ( x e. [ A ] R <-> A R x ) )
16	12, 14, 15	sylancr 412	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> A R x ) )
17		errel 6522	. . . . . . 7 |- ( R Er X -> Rel R )
18	2, 17	syl 14	. . . . . 6 |- ( ph -> Rel R )
19		brrelex2 4652	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
20	18, 19	sylan 281	. . . . 5 |- ( ( ph /\ A R B ) -> B e. _V )
21		elecg 6551	. . . . 5 |- ( ( x e. _V /\ B e. _V ) -> ( x e. [ B ] R <-> B R x ) )
22	12, 20, 21	sylancr 412	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ B ] R <-> B R x ) )
23	11, 16, 22	3bitr4d 219	. . 3 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> x e. [ B ] R ) )
24	23	eqrdv 2168	. 2 |- ( ( ph /\ A R B ) -> [ A ] R = [ B ] R )
25	2	adantr 274	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> R Er X )
26	2, 13	erref 6533	. . . . . . 7 |- ( ph -> A R A )
27	26	adantr 274	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R A )
28	13	adantr 274	. . . . . . 7 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. X )
29		elecg 6551	. . . . . . 7 |- ( ( A e. X /\ A e. X ) -> ( A e. [ A ] R <-> A R A ) )
30	28, 28, 29	syl2anc 409	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ A ] R <-> A R A ) )
31	27, 30	mpbird 166	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ A ] R )
32		simpr 109	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> [ A ] R = [ B ] R )
33	31, 32	eleqtrd 2249	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ B ] R )
34	25, 32	ereldm 6556	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. X <-> B e. X ) )
35	28, 34	mpbid 146	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B e. X )
36		elecg 6551	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A e. [ B ] R <-> B R A ) )
37	28, 35, 36	syl2anc 409	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ B ] R <-> B R A ) )
38	33, 37	mpbid 146	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B R A )
39	25, 38	ersym 6525	. 2 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R B )
40	24, 39	impbida 591	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   Rel wrel 4616   Er wer 6510   [cec 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-er 6513  df-ec 6515

This theorem is referenced by:  erth2  6558  erthi  6559  qliftfun  6595  eroveu  6604  th3qlem1  6615  enqeceq  7321  enq0eceq  7399  nnnq0lem1  7408  enreceq  7698  prsrlem1  7704