Theorem cotrg 5507

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5508 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5508. (Revised by Richard Penner, 24-Dec-2019.)

Assertion

Ref	Expression
cotrg	|- ( ( A o. B ) C_ C <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z

Proof of Theorem cotrg

Step	Hyp	Ref	Expression
1		df-co 5123	. . . 4 |- ( A o. B ) = { <. x , z >. | E. y ( x B y /\ y A z ) }
2	1	relopabi 5245	. . 3 |- Rel ( A o. B )
3		ssrel 5207	. . 3 |- ( Rel ( A o. B ) -> ( ( A o. B ) C_ C <-> A. x A. z ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) ) )
4	2, 3	ax-mp 5	. 2 |- ( ( A o. B ) C_ C <-> A. x A. z ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) )
5		vex 3203	. . . . . . . 8 |- x e. _V
6		vex 3203	. . . . . . . 8 |- z e. _V
7	5, 6	opelco 5293	. . . . . . 7 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
8		df-br 4654	. . . . . . . 8 |- ( x C z <-> <. x , z >. e. C )
9	8	bicomi 214	. . . . . . 7 |- ( <. x , z >. e. C <-> x C z )
10	7, 9	imbi12i 340	. . . . . 6 |- ( ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) <-> ( E. y ( x B y /\ y A z ) -> x C z ) )
11		19.23v 1902	. . . . . 6 |- ( A. y ( ( x B y /\ y A z ) -> x C z ) <-> ( E. y ( x B y /\ y A z ) -> x C z ) )
12	10, 11	bitr4i 267	. . . . 5 |- ( ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) <-> A. y ( ( x B y /\ y A z ) -> x C z ) )
13	12	albii 1747	. . . 4 |- ( A. z ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) <-> A. z A. y ( ( x B y /\ y A z ) -> x C z ) )
14		alcom 2037	. . . 4 |- ( A. z A. y ( ( x B y /\ y A z ) -> x C z ) <-> A. y A. z ( ( x B y /\ y A z ) -> x C z ) )
15	13, 14	bitri 264	. . 3 |- ( A. z ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) <-> A. y A. z ( ( x B y /\ y A z ) -> x C z ) )
16	15	albii 1747	. 2 |- ( A. x A. z ( <. x , z >. e. ( A o. B ) -> <. x , z >. e. C ) <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) )
17	4, 16	bitri 264	1 |- ( ( A o. B ) C_ C <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   o. ccom 5118   Rel wrel 5119

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  cotr  5508  cotr2g  13715