Theorem sscoid 32020

Description: A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)

Assertion

Ref	Expression
sscoid	|- ( A C_ ( _I o. B ) <-> ( Rel A /\ A C_ B ) )

Proof of Theorem sscoid

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5633	. . 3 |- Rel ( _I o. B )
2		relss 5206	. . 3 |- ( A C_ ( _I o. B ) -> ( Rel ( _I o. B ) -> Rel A ) )
3	1, 2	mpi 20	. 2 |- ( A C_ ( _I o. B ) -> Rel A )
4		elrel 5222	. . . . . 6 |- ( ( Rel A /\ x e. A ) -> E. y E. z x = <. y , z >. )
5		vex 3203	. . . . . . . . . . 11 |- y e. _V
6		vex 3203	. . . . . . . . . . 11 |- z e. _V
7	5, 6	brco 5292	. . . . . . . . . 10 |- ( y ( _I o. B ) z <-> E. x ( y B x /\ x _I z ) )
8		ancom 466	. . . . . . . . . . . . 13 |- ( ( y B x /\ x _I z ) <-> ( x _I z /\ y B x ) )
9	6	ideq 5274	. . . . . . . . . . . . . 14 |- ( x _I z <-> x = z )
10	9	anbi1i 731	. . . . . . . . . . . . 13 |- ( ( x _I z /\ y B x ) <-> ( x = z /\ y B x ) )
11	8, 10	bitri 264	. . . . . . . . . . . 12 |- ( ( y B x /\ x _I z ) <-> ( x = z /\ y B x ) )
12	11	exbii 1774	. . . . . . . . . . 11 |- ( E. x ( y B x /\ x _I z ) <-> E. x ( x = z /\ y B x ) )
13		breq2 4657	. . . . . . . . . . . 12 |- ( x = z -> ( y B x <-> y B z ) )
14	6, 13	ceqsexv 3242	. . . . . . . . . . 11 |- ( E. x ( x = z /\ y B x ) <-> y B z )
15	12, 14	bitri 264	. . . . . . . . . 10 |- ( E. x ( y B x /\ x _I z ) <-> y B z )
16	7, 15	bitri 264	. . . . . . . . 9 |- ( y ( _I o. B ) z <-> y B z )
17	16	a1i 11	. . . . . . . 8 |- ( x = <. y , z >. -> ( y ( _I o. B ) z <-> y B z ) )
18		eleq1 2689	. . . . . . . . 9 |- ( x = <. y , z >. -> ( x e. ( _I o. B ) <-> <. y , z >. e. ( _I o. B ) ) )
19		df-br 4654	. . . . . . . . 9 |- ( y ( _I o. B ) z <-> <. y , z >. e. ( _I o. B ) )
20	18, 19	syl6bbr 278	. . . . . . . 8 |- ( x = <. y , z >. -> ( x e. ( _I o. B ) <-> y ( _I o. B ) z ) )
21		eleq1 2689	. . . . . . . . 9 |- ( x = <. y , z >. -> ( x e. B <-> <. y , z >. e. B ) )
22		df-br 4654	. . . . . . . . 9 |- ( y B z <-> <. y , z >. e. B )
23	21, 22	syl6bbr 278	. . . . . . . 8 |- ( x = <. y , z >. -> ( x e. B <-> y B z ) )
24	17, 20, 23	3bitr4d 300	. . . . . . 7 |- ( x = <. y , z >. -> ( x e. ( _I o. B ) <-> x e. B ) )
25	24	exlimivv 1860	. . . . . 6 |- ( E. y E. z x = <. y , z >. -> ( x e. ( _I o. B ) <-> x e. B ) )
26	4, 25	syl 17	. . . . 5 |- ( ( Rel A /\ x e. A ) -> ( x e. ( _I o. B ) <-> x e. B ) )
27	26	pm5.74da 723	. . . 4 |- ( Rel A -> ( ( x e. A -> x e. ( _I o. B ) ) <-> ( x e. A -> x e. B ) ) )
28	27	albidv 1849	. . 3 |- ( Rel A -> ( A. x ( x e. A -> x e. ( _I o. B ) ) <-> A. x ( x e. A -> x e. B ) ) )
29		dfss2 3591	. . 3 |- ( A C_ ( _I o. B ) <-> A. x ( x e. A -> x e. ( _I o. B ) ) )
30		dfss2 3591	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
31	28, 29, 30	3bitr4g 303	. 2 |- ( Rel A -> ( A C_ ( _I o. B ) <-> A C_ B ) )
32	3, 31	biadan2 674	1 |- ( A C_ ( _I o. B ) <-> ( Rel A /\ A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   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-ral 2917  df-rex 2918  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-id 5024  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  dffun10  32021