Theorem vtoclr 3211

Description: Variable to class conversion of transitive relation.

Hypotheses

Ref	Expression
vtoclr.1	|- Rel R
vtoclr.2	|- ((xRy /\ yRz) -> xRz)

Assertion

Ref	Expression
vtoclr	|- (C e. D -> ((ARB /\ BRC) -> ARC))

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

Proof of Theorem vtoclr

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (C e. D -> C e. V)
2		breq1 2622	. . . . . . . 8 |- (x = A -> (xRy <-> ARy))
3	2	anbi1d 617	. . . . . . 7 |- (x = A -> ((xRy /\ yRC) <-> (ARy /\ yRC)))
4		breq1 2622	. . . . . . 7 |- (x = A -> (xRC <-> ARC))
5	3, 4	imbi12d 626	. . . . . 6 |- (x = A -> (((xRy /\ yRC) -> xRC) <-> ((ARy /\ yRC) -> ARC)))
6	5	imbi2d 612	. . . . 5 |- (x = A -> ((C e. V -> ((xRy /\ yRC) -> xRC)) <-> (C e. V -> ((ARy /\ yRC) -> ARC))))
7		breq2 2623	. . . . . . . 8 |- (y = B -> (ARy <-> ARB))
8		breq1 2622	. . . . . . . 8 |- (y = B -> (yRC <-> BRC))
9	7, 8	anbi12d 628	. . . . . . 7 |- (y = B -> ((ARy /\ yRC) <-> (ARB /\ BRC)))
10	9	imbi1d 613	. . . . . 6 |- (y = B -> (((ARy /\ yRC) -> ARC) <-> ((ARB /\ BRC) -> ARC)))
11	10	imbi2d 612	. . . . 5 |- (y = B -> ((C e. V -> ((ARy /\ yRC) -> ARC)) <-> (C e. V -> ((ARB /\ BRC) -> ARC))))
12		breq2 2623	. . . . . . . 8 |- (z = C -> (yRz <-> yRC))
13	12	anbi2d 616	. . . . . . 7 |- (z = C -> ((xRy /\ yRz) <-> (xRy /\ yRC)))
14		breq2 2623	. . . . . . 7 |- (z = C -> (xRz <-> xRC))
15	13, 14	imbi12d 626	. . . . . 6 |- (z = C -> (((xRy /\ yRz) -> xRz) <-> ((xRy /\ yRC) -> xRC)))
16		vtoclr.2	. . . . . 6 |- ((xRy /\ yRz) -> xRz)
17	15, 16	vtoclg 1847	. . . . 5 |- (C e. V -> ((xRy /\ yRC) -> xRC))
18	6, 11, 17	vtocl2g 1850	. . . 4 |- ((A e. V /\ B e. V) -> (C e. V -> ((ARB /\ BRC) -> ARC)))
19		vtoclr.1	. . . . 5 |- Rel R
20	19	brrelexi 3208	. . . 4 |- (ARB -> A e. V)
21	19	brrelexi 3208	. . . 4 |- (BRC -> B e. V)
22	18, 20, 21	syl2an 454	. . 3 |- ((ARB /\ BRC) -> (C e. V -> ((ARB /\ BRC) -> ARC)))
23	22	pm2.43b 67	. 2 |- (C e. V -> ((ARB /\ BRC) -> ARC))
24	1, 23	syl 10	1 |- (C e. D -> ((ARB /\ BRC) -> ARC))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  Rel wrel 3175

This theorem is referenced by:  vtoclrbr 3212  vtoclibr 3213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185

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