Theorem vtoclr 3206

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 1813	. 2 |- (C e. D -> C e. V)
2		breq1 2617	. . . . . . . 8 |- (x = A -> (xRy <-> ARy))
3	2	anbi1d 616	. . . . . . 7 |- (x = A -> ((xRy /\ yRC) <-> (ARy /\ yRC)))
4		breq1 2617	. . . . . . 7 |- (x = A -> (xRC <-> ARC))
5	3, 4	imbi12d 625	. . . . . 6 |- (x = A -> (((xRy /\ yRC) -> xRC) <-> ((ARy /\ yRC) -> ARC)))
6	5	imbi2d 611	. . . . 5 |- (x = A -> ((C e. V -> ((xRy /\ yRC) -> xRC)) <-> (C e. V -> ((ARy /\ yRC) -> ARC))))
7		breq2 2618	. . . . . . . 8 |- (y = B -> (ARy <-> ARB))
8		breq1 2617	. . . . . . . 8 |- (y = B -> (yRC <-> BRC))
9	7, 8	anbi12d 627	. . . . . . 7 |- (y = B -> ((ARy /\ yRC) <-> (ARB /\ BRC)))
10	9	imbi1d 612	. . . . . 6 |- (y = B -> (((ARy /\ yRC) -> ARC) <-> ((ARB /\ BRC) -> ARC)))
11	10	imbi2d 611	. . . . 5 |- (y = B -> ((C e. V -> ((ARy /\ yRC) -> ARC)) <-> (C e. V -> ((ARB /\ BRC) -> ARC))))
12		breq2 2618	. . . . . . . 8 |- (z = C -> (yRz <-> yRC))
13	12	anbi2d 615	. . . . . . 7 |- (z = C -> ((xRy /\ yRz) <-> (xRy /\ yRC)))
14		breq2 2618	. . . . . . 7 |- (z = C -> (xRz <-> xRC))
15	13, 14	imbi12d 625	. . . . . 6 |- (z = C -> (((xRy /\ yRz) -> xRz) <-> ((xRy /\ yRC) -> xRC)))
16		vtoclr.2	. . . . . 6 |- ((xRy /\ yRz) -> xRz)
17	15, 16	vtoclg 1843	. . . . 5 |- (C e. V -> ((xRy /\ yRC) -> xRC))
18	6, 11, 17	vtocl2g 1846	. . . 4 |- ((A e. V /\ B e. V) -> (C e. V -> ((ARB /\ BRC) -> ARC)))
19		vtoclr.1	. . . . 5 |- Rel R
20	19	brrelexi 3203	. . . 4 |- (ARB -> A e. V)
21	19	brrelexi 3203	. . . 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 954   e. wcel 956  Vcvv 1807   class class class wbr 2614  Rel wrel 3170

This theorem is referenced by:  vtoclrbr 3207  vtoclibr 3208

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180

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