Theorem ertr 6716

Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ersymb.1	|- ( ph -> R Er X )

Assertion

Ref	Expression
ertr	|- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )

Proof of Theorem ertr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ersymb.1	. . . . . . 7 |- ( ph -> R Er X )
2		errel 6710	. . . . . . 7 |- ( R Er X -> Rel R )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> Rel R )
4		simpr 110	. . . . . 6 |- ( ( A R B /\ B R C ) -> B R C )
5		brrelex 4766	. . . . . 6 |- ( ( Rel R /\ B R C ) -> B e. _V )
6	3, 4, 5	syl2an 289	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> B e. _V )
7		simpr 110	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> ( A R B /\ B R C ) )
8		breq2 4092	. . . . . . 7 |- ( x = B -> ( A R x <-> A R B ) )
9		breq1 4091	. . . . . . 7 |- ( x = B -> ( x R C <-> B R C ) )
10	8, 9	anbi12d 473	. . . . . 6 |- ( x = B -> ( ( A R x /\ x R C ) <-> ( A R B /\ B R C ) ) )
11	10	spcegv 2894	. . . . 5 |- ( B e. _V -> ( ( A R B /\ B R C ) -> E. x ( A R x /\ x R C ) ) )
12	6, 7, 11	sylc 62	. . . 4 |- ( ( ph /\ ( A R B /\ B R C ) ) -> E. x ( A R x /\ x R C ) )
13		simpl 109	. . . . . 6 |- ( ( A R B /\ B R C ) -> A R B )
14		brrelex 4766	. . . . . 6 |- ( ( Rel R /\ A R B ) -> A e. _V )
15	3, 13, 14	syl2an 289	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> A e. _V )
16		brrelex2 4767	. . . . . 6 |- ( ( Rel R /\ B R C ) -> C e. _V )
17	3, 4, 16	syl2an 289	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> C e. _V )
18		brcog 4897	. . . . 5 |- ( ( A e. _V /\ C e. _V ) -> ( A ( R o. R ) C <-> E. x ( A R x /\ x R C ) ) )
19	15, 17, 18	syl2anc 411	. . . 4 |- ( ( ph /\ ( A R B /\ B R C ) ) -> ( A ( R o. R ) C <-> E. x ( A R x /\ x R C ) ) )
20	12, 19	mpbird 167	. . 3 |- ( ( ph /\ ( A R B /\ B R C ) ) -> A ( R o. R ) C )
21	20	ex 115	. 2 |- ( ph -> ( ( A R B /\ B R C ) -> A ( R o. R ) C ) )
22		df-er 6701	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
23	22	simp3bi 1040	. . . . 5 |- ( R Er X -> ( `' R u. ( R o. R ) ) C_ R )
24	1, 23	syl 14	. . . 4 |- ( ph -> ( `' R u. ( R o. R ) ) C_ R )
25	24	unssbd 3385	. . 3 |- ( ph -> ( R o. R ) C_ R )
26	25	ssbrd 4131	. 2 |- ( ph -> ( A ( R o. R ) C -> A R C ) )
27	21, 26	syld 45	1 |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   u. cun 3198   C_ wss 3200   class class class wbr 4088   `'ccnv 4724   dom cdm 4725   o. ccom 4729   Rel wrel 4730   Er wer 6698

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-co 4734  df-er 6701

This theorem is referenced by:  ertrd  6717  erth  6747  iinerm  6775  entr  6957