Theorem vtoclr 4721

Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
vtoclr.1	|- Rel R
vtoclr.2	|- ( ( x R y /\ y R z ) -> x R z )

Assertion

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

Distinct variable groups:   x, y, A   y, B   x, z, C, y   x, R, y, z

Allowed substitution hints:   A( z)   B( x, z)

Proof of Theorem vtoclr

Step	Hyp	Ref	Expression
1		vtoclr.1	. . . . . 6 |- Rel R
2	1	brrelex1i 4716	. . . . 5 |- ( A R B -> A e. _V )
3	1	brrelex2i 4717	. . . . 5 |- ( A R B -> B e. _V )
4	2, 3	jca 306	. . . 4 |- ( A R B -> ( A e. _V /\ B e. _V ) )
5	1	brrelex2i 4717	. . . 4 |- ( B R C -> C e. _V )
6		breq1 4046	. . . . . . . 8 |- ( x = A -> ( x R y <-> A R y ) )
7	6	anbi1d 465	. . . . . . 7 |- ( x = A -> ( ( x R y /\ y R C ) <-> ( A R y /\ y R C ) ) )
8		breq1 4046	. . . . . . 7 |- ( x = A -> ( x R C <-> A R C ) )
9	7, 8	imbi12d 234	. . . . . 6 |- ( x = A -> ( ( ( x R y /\ y R C ) -> x R C ) <-> ( ( A R y /\ y R C ) -> A R C ) ) )
10	9	imbi2d 230	. . . . 5 |- ( x = A -> ( ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) ) <-> ( C e. _V -> ( ( A R y /\ y R C ) -> A R C ) ) ) )
11		breq2 4047	. . . . . . . 8 |- ( y = B -> ( A R y <-> A R B ) )
12		breq1 4046	. . . . . . . 8 |- ( y = B -> ( y R C <-> B R C ) )
13	11, 12	anbi12d 473	. . . . . . 7 |- ( y = B -> ( ( A R y /\ y R C ) <-> ( A R B /\ B R C ) ) )
14	13	imbi1d 231	. . . . . 6 |- ( y = B -> ( ( ( A R y /\ y R C ) -> A R C ) <-> ( ( A R B /\ B R C ) -> A R C ) ) )
15	14	imbi2d 230	. . . . 5 |- ( y = B -> ( ( C e. _V -> ( ( A R y /\ y R C ) -> A R C ) ) <-> ( C e. _V -> ( ( A R B /\ B R C ) -> A R C ) ) ) )
16		breq2 4047	. . . . . . . 8 |- ( z = C -> ( y R z <-> y R C ) )
17	16	anbi2d 464	. . . . . . 7 |- ( z = C -> ( ( x R y /\ y R z ) <-> ( x R y /\ y R C ) ) )
18		breq2 4047	. . . . . . 7 |- ( z = C -> ( x R z <-> x R C ) )
19	17, 18	imbi12d 234	. . . . . 6 |- ( z = C -> ( ( ( x R y /\ y R z ) -> x R z ) <-> ( ( x R y /\ y R C ) -> x R C ) ) )
20		vtoclr.2	. . . . . 6 |- ( ( x R y /\ y R z ) -> x R z )
21	19, 20	vtoclg 2832	. . . . 5 |- ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) )
22	10, 15, 21	vtocl2g 2836	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( C e. _V -> ( ( A R B /\ B R C ) -> A R C ) ) )
23	4, 5, 22	syl2im 38	. . 3 |- ( A R B -> ( B R C -> ( ( A R B /\ B R C ) -> A R C ) ) )
24	23	imp 124	. 2 |- ( ( A R B /\ B R C ) -> ( ( A R B /\ B R C ) -> A R C ) )
25	24	pm2.43i 49	1 |- ( ( A R B /\ B R C ) -> A R C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   class class class wbr 4043   Rel wrel 4678

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4679  df-rel 4680

This theorem is referenced by:  domtr  6862