Theorem vtoclr 4032

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 2299	. 2 |- (C e. D -> C e. _V)
2		breq1 3341	. . . . . . . 8 |- (x = A -> (xRy <-> ARy))
3	2	anbi1d 679	. . . . . . 7 |- (x = A -> ((xRy /\ yRC) <-> (ARy /\ yRC)))
4		breq1 3341	. . . . . . 7 |- (x = A -> (xRC <-> ARC))
5	3, 4	imbi12d 688	. . . . . 6 |- (x = A -> (((xRy /\ yRC) -> xRC) <-> ((ARy /\ yRC) -> ARC)))
6	5	imbi2d 674	. . . . 5 |- (x = A -> ((C e. _V -> ((xRy /\ yRC) -> xRC)) <-> (C e. _V -> ((ARy /\ yRC) -> ARC))))
7		breq2 3342	. . . . . . . 8 |- (y = B -> (ARy <-> ARB))
8		breq1 3341	. . . . . . . 8 |- (y = B -> (yRC <-> BRC))
9	7, 8	anbi12d 690	. . . . . . 7 |- (y = B -> ((ARy /\ yRC) <-> (ARB /\ BRC)))
10	9	imbi1d 675	. . . . . 6 |- (y = B -> (((ARy /\ yRC) -> ARC) <-> ((ARB /\ BRC) -> ARC)))
11	10	imbi2d 674	. . . . 5 |- (y = B -> ((C e. _V -> ((ARy /\ yRC) -> ARC)) <-> (C e. _V -> ((ARB /\ BRC) -> ARC))))
12		breq2 3342	. . . . . . . 8 |- (z = C -> (yRz <-> yRC))
13	12	anbi2d 678	. . . . . . 7 |- (z = C -> ((xRy /\ yRz) <-> (xRy /\ yRC)))
14		breq2 3342	. . . . . . 7 |- (z = C -> (xRz <-> xRC))
15	13, 14	imbi12d 688	. . . . . 6 |- (z = C -> (((xRy /\ yRz) -> xRz) <-> ((xRy /\ yRC) -> xRC)))
16		vtoclr.2	. . . . . 6 |- ((xRy /\ yRz) -> xRz)
17	15, 16	vtoclg 2346	. . . . 5 |- (C e. _V -> ((xRy /\ yRC) -> xRC))
18	6, 11, 17	vtocl2g 2349	. . . 4 |- ((A e. _V /\ B e. _V) -> (C e. _V -> ((ARB /\ BRC) -> ARC)))
19		vtoclr.1	. . . . 5 |- Rel R
20	19	brrelexi 4029	. . . 4 |- (ARB -> A e. _V)
21	19	brrelexi 4029	. . . 4 |- (BRC -> B e. _V)
22	18, 20, 21	syl2an 503	. . 3 |- ((ARB /\ BRC) -> (C e. _V -> ((ARB /\ BRC) -> ARC)))
23	22	pm2.43b 81	. 2 |- (C e. _V -> ((ARB /\ BRC) -> ARC))
24	1, 23	syl 12	1 |- (C e. D -> ((ARB /\ BRC) -> ARC))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  Rel wrel 3991

This theorem is referenced by:  vtoclrbr 4033  vtoclrbrOLD 4034  vtoclibr 4035

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001

Copyright terms: Public domain