Theorem iss2 34112

Description: A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.)

Assertion

Ref	Expression
iss2	|- ( A C_ _I <-> A = ( _I i^i ( dom A X. ran A ) ) )

Proof of Theorem iss2

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

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . . . . 9 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
2		vex 3203	. . . . . . . . . 10 |- x e. _V
3		vex 3203	. . . . . . . . . 10 |- y e. _V
4	2, 3	opeldm 5328	. . . . . . . . 9 |- ( <. x , y >. e. A -> x e. dom A )
5	1, 4	jca2 556	. . . . . . . 8 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ x e. dom A ) ) )
6	2, 3	opelrn 5357	. . . . . . . . 9 |- ( <. x , y >. e. A -> y e. ran A )
7	1, 6	jca2 556	. . . . . . . 8 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ y e. ran A ) ) )
8	5, 7	jcad 555	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> ( ( <. x , y >. e. _I /\ x e. dom A ) /\ ( <. x , y >. e. _I /\ y e. ran A ) ) ) )
9		anandi 871	. . . . . . 7 |- ( ( <. x , y >. e. _I /\ ( x e. dom A /\ y e. ran A ) ) <-> ( ( <. x , y >. e. _I /\ x e. dom A ) /\ ( <. x , y >. e. _I /\ y e. ran A ) ) )
10	8, 9	syl6ibr 242	. . . . . 6 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ ( x e. dom A /\ y e. ran A ) ) ) )
11		df-br 4654	. . . . . . . . 9 |- ( x _I y <-> <. x , y >. e. _I )
12	3	ideq 5274	. . . . . . . . 9 |- ( x _I y <-> x = y )
13	11, 12	bitr3i 266	. . . . . . . 8 |- ( <. x , y >. e. _I <-> x = y )
14	2	eldm2 5322	. . . . . . . . . . . . 13 |- ( x e. dom A <-> E. y <. x , y >. e. A )
15		opeq2 4403	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> <. x , x >. = <. x , y >. )
16	15	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
17	16	biimprcd 240	. . . . . . . . . . . . . . . 16 |- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
18	13, 17	syl5bi 232	. . . . . . . . . . . . . . 15 |- ( <. x , y >. e. A -> ( <. x , y >. e. _I -> <. x , x >. e. A ) )
19	1, 18	sylcom 30	. . . . . . . . . . . . . 14 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , x >. e. A ) )
20	19	exlimdv 1861	. . . . . . . . . . . . 13 |- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
21	14, 20	syl5bi 232	. . . . . . . . . . . 12 |- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
22	16	imbi2d 330	. . . . . . . . . . . 12 |- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
23	21, 22	syl5ibcom 235	. . . . . . . . . . 11 |- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
24	23	imp 445	. . . . . . . . . 10 |- ( ( A C_ _I /\ x = y ) -> ( x e. dom A -> <. x , y >. e. A ) )
25	24	adantrd 484	. . . . . . . . 9 |- ( ( A C_ _I /\ x = y ) -> ( ( x e. dom A /\ y e. ran A ) -> <. x , y >. e. A ) )
26	25	ex 450	. . . . . . . 8 |- ( A C_ _I -> ( x = y -> ( ( x e. dom A /\ y e. ran A ) -> <. x , y >. e. A ) ) )
27	13, 26	syl5bi 232	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. _I -> ( ( x e. dom A /\ y e. ran A ) -> <. x , y >. e. A ) ) )
28	27	impd 447	. . . . . 6 |- ( A C_ _I -> ( ( <. x , y >. e. _I /\ ( x e. dom A /\ y e. ran A ) ) -> <. x , y >. e. A ) )
29	10, 28	impbid 202	. . . . 5 |- ( A C_ _I -> ( <. x , y >. e. A <-> ( <. x , y >. e. _I /\ ( x e. dom A /\ y e. ran A ) ) ) )
30		opelinxp 34111	. . . . . 6 |- ( <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) <-> ( ( x e. dom A /\ y e. ran A ) /\ <. x , y >. e. _I ) )
31	30	biancom 33994	. . . . 5 |- ( <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) <-> ( <. x , y >. e. _I /\ ( x e. dom A /\ y e. ran A ) ) )
32	29, 31	syl6bbr 278	. . . 4 |- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) ) )
33	32	alrimivv 1856	. . 3 |- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) ) )
34		reli 5249	. . . . 5 |- Rel _I
35		relss 5206	. . . . 5 |- ( A C_ _I -> ( Rel _I -> Rel A ) )
36	34, 35	mpi 20	. . . 4 |- ( A C_ _I -> Rel A )
37		relinxp 34069	. . . 4 |- Rel ( _I i^i ( dom A X. ran A ) )
38		eqrel 5209	. . . 4 |- ( ( Rel A /\ Rel ( _I i^i ( dom A X. ran A ) ) ) -> ( A = ( _I i^i ( dom A X. ran A ) ) <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) ) ) )
39	36, 37, 38	sylancl 694	. . 3 |- ( A C_ _I -> ( A = ( _I i^i ( dom A X. ran A ) ) <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I i^i ( dom A X. ran A ) ) ) ) )
40	33, 39	mpbird 247	. 2 |- ( A C_ _I -> A = ( _I i^i ( dom A X. ran A ) ) )
41		inss1 3833	. . 3 |- ( _I i^i ( dom A X. ran A ) ) C_ _I
42		sseq1 3626	. . 3 |- ( A = ( _I i^i ( dom A X. ran A ) ) -> ( A C_ _I <-> ( _I i^i ( dom A X. ran A ) ) C_ _I ) )
43	41, 42	mpbiri 248	. 2 |- ( A = ( _I i^i ( dom A X. ran A ) ) -> A C_ _I )
44	40, 43	impbii 199	1 |- ( A C_ _I <-> A = ( _I i^i ( dom A X. ran A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   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-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)