Theorem ertr 6760

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 6754	. . . . . . 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 4772	. . . . . 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 4097	. . . . . . 7 |- ( x = B -> ( A R x <-> A R B ) )
9		breq1 4096	. . . . . . 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 2895	. . . . 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 4772	. . . . . 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 4773	. . . . . 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 4903	. . . . 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 6745	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
23	22	simp3bi 1041	. . . . 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 3387	. . 3 |- ( ph -> ( R o. R ) C_ R )
26	25	ssbrd 4136	. 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 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   u. cun 3199   C_ wss 3201   class class class wbr 4093   `'ccnv 4730   dom cdm 4731   o. ccom 4735   Rel wrel 4736   Er wer 6742

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-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-co 4740  df-er 6745

This theorem is referenced by:  ertrd  6761  erth  6791  iinerm  6819  entr  7001