Theorem vtoclr 4774

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 4769	. . . . 5 |- ( A R B -> A e. _V )
3	1	brrelex2i 4770	. . . . 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 4770	. . . 4 |- ( B R C -> C e. _V )
6		breq1 4091	. . . . . . . 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 4091	. . . . . . 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 4092	. . . . . . . 8 |- ( y = B -> ( A R y <-> A R B ) )
12		breq1 4091	. . . . . . . 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 4092	. . . . . . . 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 4092	. . . . . . 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 2864	. . . . 5 |- ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) )
22	10, 15, 21	vtocl2g 2868	. . . 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 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732

This theorem is referenced by:  domtr  6958