Theorem ertr 6516

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 6510	. . . . . . 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 4644	. . . . . 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 3986	. . . . . . 7 |- ( x = B -> ( A R x <-> A R B ) )
9		breq1 3985	. . . . . . 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 2814	. . . . 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 4644	. . . . . 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 4645	. . . . . 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 4771	. . . . 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 6501	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
23	22	simp3bi 1004	. . . . 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 3300	. . 3 |- ( ph -> ( R o. R ) C_ R )
26	25	ssbrd 4025	. 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 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116   class class class wbr 3982   `'ccnv 4603   dom cdm 4604   o. ccom 4608   Rel wrel 4609   Er wer 6498

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-co 4613  df-er 6501

This theorem is referenced by:  ertrd  6517  erth  6545  iinerm  6573  entr  6750