Theorem cotrintab 37921

Description: The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)

Hypothesis

Ref	Expression
cotrintab.min	|- ( ph -> ( x o. x ) C_ x )

Assertion

Ref	Expression
cotrintab	|- ( |^| { x | ph } o. |^| { x | ph } ) C_ |^| { x | ph }

Proof of Theorem cotrintab

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cotr 5508	. 2 |- ( ( |^| { x | ph } o. |^| { x | ph } ) C_ |^| { x | ph } <-> A. u A. w A. v ( ( u |^| { x | ph } w /\ w |^| { x | ph } v ) -> u |^| { x | ph } v ) )
2		pm3.43 906	. . . . . 6 |- ( ( ( ph -> u x w ) /\ ( ph -> w x v ) ) -> ( ph -> ( u x w /\ w x v ) ) )
3		cotrintab.min	. . . . . . 7 |- ( ph -> ( x o. x ) C_ x )
4		cotr 5508	. . . . . . . 8 |- ( ( x o. x ) C_ x <-> A. u A. w A. v ( ( u x w /\ w x v ) -> u x v ) )
5	4	biimpi 206	. . . . . . 7 |- ( ( x o. x ) C_ x -> A. u A. w A. v ( ( u x w /\ w x v ) -> u x v ) )
6		2sp 2056	. . . . . . . 8 |- ( A. w A. v ( ( u x w /\ w x v ) -> u x v ) -> ( ( u x w /\ w x v ) -> u x v ) )
7	6	sps 2055	. . . . . . 7 |- ( A. u A. w A. v ( ( u x w /\ w x v ) -> u x v ) -> ( ( u x w /\ w x v ) -> u x v ) )
8	3, 5, 7	3syl 18	. . . . . 6 |- ( ph -> ( ( u x w /\ w x v ) -> u x v ) )
9	2, 8	sylcom 30	. . . . 5 |- ( ( ( ph -> u x w ) /\ ( ph -> w x v ) ) -> ( ph -> u x v ) )
10	9	alanimi 1744	. . . 4 |- ( ( A. x ( ph -> u x w ) /\ A. x ( ph -> w x v ) ) -> A. x ( ph -> u x v ) )
11		opex 4932	. . . . . . 7 |- <. u , w >. e. _V
12	11	elintab 4487	. . . . . 6 |- ( <. u , w >. e. |^| { x | ph } <-> A. x ( ph -> <. u , w >. e. x ) )
13		df-br 4654	. . . . . 6 |- ( u |^| { x | ph } w <-> <. u , w >. e. |^| { x | ph } )
14		df-br 4654	. . . . . . . 8 |- ( u x w <-> <. u , w >. e. x )
15	14	imbi2i 326	. . . . . . 7 |- ( ( ph -> u x w ) <-> ( ph -> <. u , w >. e. x ) )
16	15	albii 1747	. . . . . 6 |- ( A. x ( ph -> u x w ) <-> A. x ( ph -> <. u , w >. e. x ) )
17	12, 13, 16	3bitr4i 292	. . . . 5 |- ( u |^| { x | ph } w <-> A. x ( ph -> u x w ) )
18		opex 4932	. . . . . . 7 |- <. w , v >. e. _V
19	18	elintab 4487	. . . . . 6 |- ( <. w , v >. e. |^| { x | ph } <-> A. x ( ph -> <. w , v >. e. x ) )
20		df-br 4654	. . . . . 6 |- ( w |^| { x | ph } v <-> <. w , v >. e. |^| { x | ph } )
21		df-br 4654	. . . . . . . 8 |- ( w x v <-> <. w , v >. e. x )
22	21	imbi2i 326	. . . . . . 7 |- ( ( ph -> w x v ) <-> ( ph -> <. w , v >. e. x ) )
23	22	albii 1747	. . . . . 6 |- ( A. x ( ph -> w x v ) <-> A. x ( ph -> <. w , v >. e. x ) )
24	19, 20, 23	3bitr4i 292	. . . . 5 |- ( w |^| { x | ph } v <-> A. x ( ph -> w x v ) )
25	17, 24	anbi12i 733	. . . 4 |- ( ( u |^| { x | ph } w /\ w |^| { x | ph } v ) <-> ( A. x ( ph -> u x w ) /\ A. x ( ph -> w x v ) ) )
26		opex 4932	. . . . . 6 |- <. u , v >. e. _V
27	26	elintab 4487	. . . . 5 |- ( <. u , v >. e. |^| { x | ph } <-> A. x ( ph -> <. u , v >. e. x ) )
28		df-br 4654	. . . . 5 |- ( u |^| { x | ph } v <-> <. u , v >. e. |^| { x | ph } )
29		df-br 4654	. . . . . . 7 |- ( u x v <-> <. u , v >. e. x )
30	29	imbi2i 326	. . . . . 6 |- ( ( ph -> u x v ) <-> ( ph -> <. u , v >. e. x ) )
31	30	albii 1747	. . . . 5 |- ( A. x ( ph -> u x v ) <-> A. x ( ph -> <. u , v >. e. x ) )
32	27, 28, 31	3bitr4i 292	. . . 4 |- ( u |^| { x | ph } v <-> A. x ( ph -> u x v ) )
33	10, 25, 32	3imtr4i 281	. . 3 |- ( ( u |^| { x | ph } w /\ w |^| { x | ph } v ) -> u |^| { x | ph } v )
34	33	gen2 1723	. 2 |- A. w A. v ( ( u |^| { x | ph } w /\ w |^| { x | ph } v ) -> u |^| { x | ph } v )
35	1, 34	mpgbir 1726	1 |- ( |^| { x | ph } o. |^| { x | ph } ) C_ |^| { x | ph }

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   C_ wss 3574   <.cop 4183   |^|cint 4475   class class class wbr 4653   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  dfrtrcl5  37936