Theorem vtoclr 4545

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 4540	. . . . 5 |- ( A R B -> A e. _V )
3	1	brrelex2i 4541	. . . . 5 |- ( A R B -> B e. _V )
4	2, 3	jca 302	. . . 4 |- ( A R B -> ( A e. _V /\ B e. _V ) )
5	1	brrelex2i 4541	. . . 4 |- ( B R C -> C e. _V )
6		breq1 3896	. . . . . . . 8 |- ( x = A -> ( x R y <-> A R y ) )
7	6	anbi1d 458	. . . . . . 7 |- ( x = A -> ( ( x R y /\ y R C ) <-> ( A R y /\ y R C ) ) )
8		breq1 3896	. . . . . . 7 |- ( x = A -> ( x R C <-> A R C ) )
9	7, 8	imbi12d 233	. . . . . 6 |- ( x = A -> ( ( ( x R y /\ y R C ) -> x R C ) <-> ( ( A R y /\ y R C ) -> A R C ) ) )
10	9	imbi2d 229	. . . . 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 3897	. . . . . . . 8 |- ( y = B -> ( A R y <-> A R B ) )
12		breq1 3896	. . . . . . . 8 |- ( y = B -> ( y R C <-> B R C ) )
13	11, 12	anbi12d 462	. . . . . . 7 |- ( y = B -> ( ( A R y /\ y R C ) <-> ( A R B /\ B R C ) ) )
14	13	imbi1d 230	. . . . . 6 |- ( y = B -> ( ( ( A R y /\ y R C ) -> A R C ) <-> ( ( A R B /\ B R C ) -> A R C ) ) )
15	14	imbi2d 229	. . . . 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 3897	. . . . . . . 8 |- ( z = C -> ( y R z <-> y R C ) )
17	16	anbi2d 457	. . . . . . 7 |- ( z = C -> ( ( x R y /\ y R z ) <-> ( x R y /\ y R C ) ) )
18		breq2 3897	. . . . . . 7 |- ( z = C -> ( x R z <-> x R C ) )
19	17, 18	imbi12d 233	. . . . . 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 2715	. . . . 5 |- ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) )
22	10, 15, 21	vtocl2g 2719	. . . 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 123	. 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 103   = wceq 1312   e. wcel 1461   _Vcvv 2655   class class class wbr 3893   Rel wrel 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504

This theorem is referenced by:  domtr  6631