Theorem erth 6791

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 109	. . . . . . 7 |- ( ( ph /\ A R B ) -> ph )
2		erth.1	. . . . . . . . 9 |- ( ph -> R Er X )
3	2	ersymb 6759	. . . . . . . 8 |- ( ph -> ( A R B <-> B R A ) )
4	3	biimpa 296	. . . . . . 7 |- ( ( ph /\ A R B ) -> B R A )
5	1, 4	jca 306	. . . . . 6 |- ( ( ph /\ A R B ) -> ( ph /\ B R A ) )
6	2	ertr 6760	. . . . . . 7 |- ( ph -> ( ( B R A /\ A R x ) -> B R x ) )
7	6	impl 380	. . . . . 6 |- ( ( ( ph /\ B R A ) /\ A R x ) -> B R x )
8	5, 7	sylan 283	. . . . 5 |- ( ( ( ph /\ A R B ) /\ A R x ) -> B R x )
9	2	ertr 6760	. . . . . 6 |- ( ph -> ( ( A R B /\ B R x ) -> A R x ) )
10	9	impl 380	. . . . 5 |- ( ( ( ph /\ A R B ) /\ B R x ) -> A R x )
11	8, 10	impbida 600	. . . 4 |- ( ( ph /\ A R B ) -> ( A R x <-> B R x ) )
12		vex 2806	. . . . 5 |- x e. _V
13		erth.2	. . . . . 6 |- ( ph -> A e. X )
14	13	adantr 276	. . . . 5 |- ( ( ph /\ A R B ) -> A e. X )
15		elecg 6785	. . . . 5 |- ( ( x e. _V /\ A e. X ) -> ( x e. [ A ] R <-> A R x ) )
16	12, 14, 15	sylancr 414	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> A R x ) )
17		errel 6754	. . . . . . 7 |- ( R Er X -> Rel R )
18	2, 17	syl 14	. . . . . 6 |- ( ph -> Rel R )
19		brrelex2 4773	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
20	18, 19	sylan 283	. . . . 5 |- ( ( ph /\ A R B ) -> B e. _V )
21		elecg 6785	. . . . 5 |- ( ( x e. _V /\ B e. _V ) -> ( x e. [ B ] R <-> B R x ) )
22	12, 20, 21	sylancr 414	. . . 4 |- ( ( ph /\ A R B ) -> ( x e. [ B ] R <-> B R x ) )
23	11, 16, 22	3bitr4d 220	. . 3 |- ( ( ph /\ A R B ) -> ( x e. [ A ] R <-> x e. [ B ] R ) )
24	23	eqrdv 2229	. 2 |- ( ( ph /\ A R B ) -> [ A ] R = [ B ] R )
25	2	adantr 276	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> R Er X )
26	2, 13	erref 6765	. . . . . . 7 |- ( ph -> A R A )
27	26	adantr 276	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R A )
28	13	adantr 276	. . . . . . 7 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. X )
29		elecg 6785	. . . . . . 7 |- ( ( A e. X /\ A e. X ) -> ( A e. [ A ] R <-> A R A ) )
30	28, 28, 29	syl2anc 411	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ A ] R <-> A R A ) )
31	27, 30	mpbird 167	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ A ] R )
32		simpr 110	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> [ A ] R = [ B ] R )
33	31, 32	eleqtrd 2310	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A e. [ B ] R )
34	25, 32	ereldm 6790	. . . . . 6 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. X <-> B e. X ) )
35	28, 34	mpbid 147	. . . . 5 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B e. X )
36		elecg 6785	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A e. [ B ] R <-> B R A ) )
37	28, 35, 36	syl2anc 411	. . . 4 |- ( ( ph /\ [ A ] R = [ B ] R ) -> ( A e. [ B ] R <-> B R A ) )
38	33, 37	mpbid 147	. . 3 |- ( ( ph /\ [ A ] R = [ B ] R ) -> B R A )
39	25, 38	ersym 6757	. 2 |- ( ( ph /\ [ A ] R = [ B ] R ) -> A R B )
40	24, 39	impbida 600	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   Rel wrel 4736   Er wer 6742   [cec 6743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-er 6745  df-ec 6747

This theorem is referenced by:  erth2  6792  erthi  6793  qliftfun  6829  eroveu  6838  th3qlem1  6849  enqeceq  7639  enq0eceq  7717  nnnq0lem1  7726  enreceq  8016  prsrlem1  8022  ercpbllemg  13493  eqg0el  13896  qusecsub  13998  zndvds  14745