Theorem ertr 6812

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 6806	. . . . . . 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 4810	. . . . . 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 4129	. . . . . . 7 |- ( x = B -> ( A R x <-> A R B ) )
9		breq1 4128	. . . . . . 7 |- ( x = B -> ( x R C <-> B R C ) )
10	8, 9	anbi12d 477	. . . . . 6 |- ( x = B -> ( ( A R x /\ x R C ) <-> ( A R B /\ B R C ) ) )
11	10	spcegv 2913	. . . . 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 4810	. . . . . 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 4811	. . . . . 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 4942	. . . . 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 415	. . . 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 6797	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
23	22	simp3bi 1045	. . . . 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 3407	. . 3 |- ( ph -> ( R o. R ) C_ R )
26	25	ssbrd 4168	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   class class class wbr 4125   `'ccnv 4768   dom cdm 4769   o. ccom 4773   Rel wrel 4774   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-co 4778  df-er 6797

This theorem is referenced by:  ertrd  6813  erth  6843  iinerm  6871  entr  7061