Theorem ertr 6452

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 6446	. . . . . . 7 |- ( R Er X -> Rel R )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> Rel R )
4		simpr 109	. . . . . 6 |- ( ( A R B /\ B R C ) -> B R C )
5		brrelex 4587	. . . . . 6 |- ( ( Rel R /\ B R C ) -> B e. _V )
6	3, 4, 5	syl2an 287	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> B e. _V )
7		simpr 109	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> ( A R B /\ B R C ) )
8		breq2 3941	. . . . . . 7 |- ( x = B -> ( A R x <-> A R B ) )
9		breq1 3940	. . . . . . 7 |- ( x = B -> ( x R C <-> B R C ) )
10	8, 9	anbi12d 465	. . . . . 6 |- ( x = B -> ( ( A R x /\ x R C ) <-> ( A R B /\ B R C ) ) )
11	10	spcegv 2777	. . . . 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 108	. . . . . 6 |- ( ( A R B /\ B R C ) -> A R B )
14		brrelex 4587	. . . . . 6 |- ( ( Rel R /\ A R B ) -> A e. _V )
15	3, 13, 14	syl2an 287	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> A e. _V )
16		brrelex2 4588	. . . . . 6 |- ( ( Rel R /\ B R C ) -> C e. _V )
17	3, 4, 16	syl2an 287	. . . . 5 |- ( ( ph /\ ( A R B /\ B R C ) ) -> C e. _V )
18		brcog 4714	. . . . 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 409	. . . 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 166	. . 3 |- ( ( ph /\ ( A R B /\ B R C ) ) -> A ( R o. R ) C )
21	20	ex 114	. 2 |- ( ph -> ( ( A R B /\ B R C ) -> A ( R o. R ) C ) )
22		df-er 6437	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
23	22	simp3bi 999	. . . . 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 3259	. . 3 |- ( ph -> ( R o. R ) C_ R )
26	25	ssbrd 3979	. 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 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   u. cun 3074   C_ wss 3076   class class class wbr 3937   `'ccnv 4546   dom cdm 4547   o. ccom 4551   Rel wrel 4552   Er wer 6434

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-co 4556  df-er 6437

This theorem is referenced by:  ertrd  6453  erth  6481  iinerm  6509  entr  6686