Theorem vtoclr 5164

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 5158	. . . . 5 |- ( A R B -> A e. _V )
3	1	brrelex2i 5159	. . . . 5 |- ( A R B -> B e. _V )
4	2, 3	jca 554	. . . 4 |- ( A R B -> ( A e. _V /\ B e. _V ) )
5	1	brrelex2i 5159	. . . 4 |- ( B R C -> C e. _V )
6		breq1 4656	. . . . . . . 8 |- ( x = A -> ( x R y <-> A R y ) )
7	6	anbi1d 741	. . . . . . 7 |- ( x = A -> ( ( x R y /\ y R C ) <-> ( A R y /\ y R C ) ) )
8		breq1 4656	. . . . . . 7 |- ( x = A -> ( x R C <-> A R C ) )
9	7, 8	imbi12d 334	. . . . . 6 |- ( x = A -> ( ( ( x R y /\ y R C ) -> x R C ) <-> ( ( A R y /\ y R C ) -> A R C ) ) )
10	9	imbi2d 330	. . . . 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 4657	. . . . . . . 8 |- ( y = B -> ( A R y <-> A R B ) )
12		breq1 4656	. . . . . . . 8 |- ( y = B -> ( y R C <-> B R C ) )
13	11, 12	anbi12d 747	. . . . . . 7 |- ( y = B -> ( ( A R y /\ y R C ) <-> ( A R B /\ B R C ) ) )
14	13	imbi1d 331	. . . . . 6 |- ( y = B -> ( ( ( A R y /\ y R C ) -> A R C ) <-> ( ( A R B /\ B R C ) -> A R C ) ) )
15	14	imbi2d 330	. . . . 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 4657	. . . . . . . 8 |- ( z = C -> ( y R z <-> y R C ) )
17	16	anbi2d 740	. . . . . . 7 |- ( z = C -> ( ( x R y /\ y R z ) <-> ( x R y /\ y R C ) ) )
18		breq2 4657	. . . . . . 7 |- ( z = C -> ( x R z <-> x R C ) )
19	17, 18	imbi12d 334	. . . . . 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 3266	. . . . 5 |- ( C e. _V -> ( ( x R y /\ y R C ) -> x R C ) )
22	10, 15, 21	vtocl2g 3270	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( C e. _V -> ( ( A R B /\ B R C ) -> A R C ) ) )
23	4, 5, 22	syl2im 40	. . 3 |- ( A R B -> ( B R C -> ( ( A R B /\ B R C ) -> A R C ) ) )
24	23	imp 445	. 2 |- ( ( A R B /\ B R C ) -> ( ( A R B /\ B R C ) -> A R C ) )
25	24	pm2.43i 52	1 |- ( ( A R B /\ B R C ) -> A R C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  domtr  8009