Theorem inttr 14384

Description: The intersection of two transitive classes is transitive.

Assertion

Ref	Expression
inttr	|- (((R o. R) C_ R /\ (S o. S) C_ S) -> ((R i^i S) o. (R i^i S)) C_ (R i^i S))

Proof of Theorem inttr

Step	Hyp	Ref	Expression
1		cotr 4302	. . . 4 |- ((R o. R) C_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
2		cotr 4302	. . . . . 6 |- ((S o. S) C_ S <-> A.xA.yA.z((xSy /\ ySz) -> xSz))
3		19.26 1416	. . . . . . . 8 |- (A.x(A.yA.z((xSy /\ ySz) -> xSz) /\ A.yA.z((xRy /\ yRz) -> xRz)) <-> (A.xA.yA.z((xSy /\ ySz) -> xSz) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)))
4		19.26 1416	. . . . . . . . . 10 |- (A.y(A.z((xSy /\ ySz) -> xSz) /\ A.z((xRy /\ yRz) -> xRz)) <-> (A.yA.z((xSy /\ ySz) -> xSz) /\ A.yA.z((xRy /\ yRz) -> xRz)))
5		19.26 1416	. . . . . . . . . . . 12 |- (A.z(((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) <-> (A.z((xSy /\ ySz) -> xSz) /\ A.z((xRy /\ yRz) -> xRz)))
6		simpr 350	. . . . . . . . . . . . . . . . . . 19 |- ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> ((xRy /\ yRz) -> xRz))
7		simpl 346	. . . . . . . . . . . . . . . . . . 19 |- ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> ((xSy /\ ySz) -> xSz))
8	6, 7	anim12d 617	. . . . . . . . . . . . . . . . . 18 |- ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> (((xRy /\ yRz) /\ (xSy /\ ySz)) -> (xRz /\ xSz)))
9	8	com12 14	. . . . . . . . . . . . . . . . 17 |- (((xRy /\ yRz) /\ (xSy /\ ySz)) -> ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> (xRz /\ xSz)))
10	9	an4s 566	. . . . . . . . . . . . . . . 16 |- (((xRy /\ xSy) /\ (yRz /\ ySz)) -> ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> (xRz /\ xSz)))
11		brin 3388	. . . . . . . . . . . . . . . 16 |- (x(R i^i S)y <-> (xRy /\ xSy))
12		brin 3388	. . . . . . . . . . . . . . . 16 |- (y(R i^i S)z <-> (yRz /\ ySz))
13	10, 11, 12	syl2anb 504	. . . . . . . . . . . . . . 15 |- ((x(R i^i S)y /\ y(R i^i S)z) -> ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> (xRz /\ xSz)))
14	13	com12 14	. . . . . . . . . . . . . 14 |- ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> ((x(R i^i S)y /\ y(R i^i S)z) -> (xRz /\ xSz)))
15		brin 3388	. . . . . . . . . . . . . 14 |- (x(R i^i S)z <-> (xRz /\ xSz))
16	14, 15	syl6ibr 230	. . . . . . . . . . . . 13 |- ((((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> ((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
17	16	alimi 1338	. . . . . . . . . . . 12 |- (A.z(((xSy /\ ySz) -> xSz) /\ ((xRy /\ yRz) -> xRz)) -> A.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
18	5, 17	sylbir 218	. . . . . . . . . . 11 |- ((A.z((xSy /\ ySz) -> xSz) /\ A.z((xRy /\ yRz) -> xRz)) -> A.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
19	18	alimi 1338	. . . . . . . . . 10 |- (A.y(A.z((xSy /\ ySz) -> xSz) /\ A.z((xRy /\ yRz) -> xRz)) -> A.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
20	4, 19	sylbir 218	. . . . . . . . 9 |- ((A.yA.z((xSy /\ ySz) -> xSz) /\ A.yA.z((xRy /\ yRz) -> xRz)) -> A.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
21	20	alimi 1338	. . . . . . . 8 |- (A.x(A.yA.z((xSy /\ ySz) -> xSz) /\ A.yA.z((xRy /\ yRz) -> xRz)) -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
22	3, 21	sylbir 218	. . . . . . 7 |- ((A.xA.yA.z((xSy /\ ySz) -> xSz) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)) -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
23	22	ex 402	. . . . . 6 |- (A.xA.yA.z((xSy /\ ySz) -> xSz) -> (A.xA.yA.z((xRy /\ yRz) -> xRz) -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z)))
24	2, 23	sylbi 216	. . . . 5 |- ((S o. S) C_ S -> (A.xA.yA.z((xRy /\ yRz) -> xRz) -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z)))
25	24	com12 14	. . . 4 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) -> ((S o. S) C_ S -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z)))
26	1, 25	sylbi 216	. . 3 |- ((R o. R) C_ R -> ((S o. S) C_ S -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z)))
27	26	imp 377	. 2 |- (((R o. R) C_ R /\ (S o. S) C_ S) -> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
28		cotr 4302	. 2 |- (((R i^i S) o. (R i^i S)) C_ (R i^i S) <-> A.xA.yA.z((x(R i^i S)y /\ y(R i^i S)z) -> x(R i^i S)z))
29	27, 28	sylibr 217	1 |- (((R o. R) C_ R /\ (S o. S) C_ S) -> ((R i^i S) o. (R i^i S)) C_ (R i^i S))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   i^i cin 2592   C_ wss 2593   class class class wbr 3338   o. ccom 3990

This theorem is referenced by:  tricptr 14385  int2pre 14578

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-eu 1775  df-mo 1776  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  df-co 4003

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