Theorem vtoclr 4388

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	brrelexi 4384	. . . . 5 |- ( A R B -> A e. _V )
3	1	brrelex2i 4385	. . . . 5 |- ( A R B -> B e. _V )
4	2, 3	jca 290	. . . 4 |- ( A R B -> ( A e. _V /\ B e. _V ) )
5	1	brrelex2i 4385	. . . 4 |- ( B R C -> C e. _V )
6		breq1 3767	. . . . . . . 8 |- ( x = A -> ( x R y <-> A R y ) )
7	6	anbi1d 438	. . . . . . 7 |- ( x = A -> ( ( x R y /\ y R C ) <-> ( A R y /\ y R C ) ) )
8		breq1 3767	. . . . . . 7 |- ( x = A -> ( x R C <-> A R C ) )
9	7, 8	imbi12d 223	. . . . . 6 |- ( x = A -> ( ( ( x R y /\ y R C ) -> x R C ) <-> ( ( A R y /\ y R C ) -> A R C ) ) )
10	9	imbi2d 219	. . . . 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 3768	. . . . . . . 8 |- ( y = B -> ( A R y <-> A R B ) )
12		breq1 3767	. . . . . . . 8 |- ( y = B -> ( y R C <-> B R C ) )
13	11, 12	anbi12d 442	. . . . . . 7 |- ( y = B -> ( ( A R y /\ y R C ) <-> ( A R B /\ B R C ) ) )
14	13	imbi1d 220	. . . . . 6 |- ( y = B -> ( ( ( A R y /\ y R C ) -> A R C ) <-> ( ( A R B /\ B R C ) -> A R C ) ) )
15	14	imbi2d 219	. . . . 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 3768	. . . . . . . 8 |- ( z = C -> ( y R z <-> y R C ) )
17	16	anbi2d 437	. . . . . . 7 |- ( z = C -> ( ( x R y /\ y R z ) <-> ( x R y /\ y R C ) ) )
18		breq2 3768	. . . . . . 7 |- ( z = C -> ( x R z <-> x R C ) )
19	17, 18	imbi12d 223	. . . . . 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 2613	. . . . 5 |- ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) )
22	10, 15, 21	vtocl2g 2617	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( C e. _V -> ( ( A R B /\ B R C ) -> A R C ) ) )
23	4, 5, 22	syl2im 34	. . 3 |- ( A R B -> ( B R C -> ( ( A R B /\ B R C ) -> A R C ) ) )
24	23	imp 115	. 2 |- ( ( A R B /\ B R C ) -> ( ( A R B /\ B R C ) -> A R C ) )
25	24	pm2.43i 43	1 |- ( ( A R B /\ B R C ) -> A R C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352

This theorem is referenced by:  domtr  6265