Theorem vtoclr 2449

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 group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,R,y,z

Proof of Theorem vtoclr

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (C e. D -> C e. V)
2		breq1 2065	. . . . . . . 8 |- (x = A -> (xRy <-> ARy))
3	2	anbi1d 469	. . . . . . 7 |- (x = A -> ((xRy /\ yRC) <-> (ARy /\ yRC)))
4		breq1 2065	. . . . . . 7 |- (x = A -> (xRC <-> ARC))
5	3, 4	imbi12d 474	. . . . . 6 |- (x = A -> (((xRy /\ yRC) -> xRC) <-> ((ARy /\ yRC) -> ARC)))
6	5	imbi2d 464	. . . . 5 |- (x = A -> ((C e. V -> ((xRy /\ yRC) -> xRC)) <-> (C e. V -> ((ARy /\ yRC) -> ARC))))
7		breq2 2066	. . . . . . . 8 |- (y = B -> (ARy <-> ARB))
8		breq1 2065	. . . . . . . 8 |- (y = B -> (yRC <-> BRC))
9	7, 8	anbi12d 476	. . . . . . 7 |- (y = B -> ((ARy /\ yRC) <-> (ARB /\ BRC)))
10	9	imbi1d 465	. . . . . 6 |- (y = B -> (((ARy /\ yRC) -> ARC) <-> ((ARB /\ BRC) -> ARC)))
11	10	imbi2d 464	. . . . 5 |- (y = B -> ((C e. V -> ((ARy /\ yRC) -> ARC)) <-> (C e. V -> ((ARB /\ BRC) -> ARC))))
12		breq2 2066	. . . . . . . 8 |- (z = C -> (yRz <-> yRC))
13	12	anbi2d 468	. . . . . . 7 |- (z = C -> ((xRy /\ yRz) <-> (xRy /\ yRC)))
14		breq2 2066	. . . . . . 7 |- (z = C -> (xRz <-> xRC))
15	13, 14	imbi12d 474	. . . . . 6 |- (z = C -> (((xRy /\ yRz) -> xRz) <-> ((xRy /\ yRC) -> xRC)))
16		vtoclr.2	. . . . . 6 |- ((xRy /\ yRz) -> xRz)
17	15, 16	vtoclg 1383	. . . . 5 |- (C e. V -> ((xRy /\ yRC) -> xRC))
18	6, 11, 17	vtocl2g 1386	. . . 4 |- ((A e. V /\ B e. V) -> (C e. V -> ((ARB /\ BRC) -> ARC)))
19		vtoclr.1	. . . . 5 |- Rel R
20	19	brrelexi 2447	. . . 4 |- (ARB -> A e. V)
21	19	brrelexi 2447	. . . 4 |- (BRC -> B e. V)
22	18, 20, 21	syl2an 349	. . 3 |- ((ARB /\ BRC) -> (C e. V -> ((ARB /\ BRC) -> ARC)))
23	22	pm2.43b 61	. 2 |- (C e. V -> ((ARB /\ BRC) -> ARC))
24	1, 23	syl 12	1 |- (C e. D -> ((ARB /\ BRC) -> ARC))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  Rel wrel 2415

This theorem is referenced by:  vtoclrbr 2450  vtoclibr 2451

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425

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