Theorem bj-elid 33085

Description: Characterization of the elements of _I. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-elid	|- ( A e. _I <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) )

Proof of Theorem bj-elid

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

Step	Hyp	Ref	Expression
1		reli 5249	. . . . 5 |- Rel _I
2		df-rel 5121	. . . . 5 |- ( Rel _I <-> _I C_ ( _V X. _V ) )
3	1, 2	mpbi 220	. . . 4 |- _I C_ ( _V X. _V )
4	3	sseli 3599	. . 3 |- ( A e. _I -> A e. ( _V X. _V ) )
5		1st2nd2 7205	. . . . . . 7 |- ( A e. ( _V X. _V ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
6	4, 5	syl 17	. . . . . 6 |- ( A e. _I -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
7	6	eleq1d 2686	. . . . 5 |- ( A e. _I -> ( A e. _I <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I ) )
8	7	ibi 256	. . . 4 |- ( A e. _I -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I )
9		df-id 5024	. . . . . 6 |- _I = { <. x , y >. | x = y }
10	9	eleq2i 2693	. . . . 5 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | x = y } )
11		fvex 6201	. . . . . 6 |- ( 1st ` A ) e. _V
12		fvex 6201	. . . . . 6 |- ( 2nd ` A ) e. _V
13		eqeq12 2635	. . . . . 6 |- ( ( x = ( 1st ` A ) /\ y = ( 2nd ` A ) ) -> ( x = y <-> ( 1st ` A ) = ( 2nd ` A ) ) )
14	11, 12, 13	opelopaba 4991	. . . . 5 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | x = y } <-> ( 1st ` A ) = ( 2nd ` A ) )
15	10, 14	bitri 264	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I <-> ( 1st ` A ) = ( 2nd ` A ) )
16	8, 15	sylib 208	. . 3 |- ( A e. _I -> ( 1st ` A ) = ( 2nd ` A ) )
17	4, 16	jca 554	. 2 |- ( A e. _I -> ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) )
18	5	eleq1d 2686	. . . . 5 |- ( A e. ( _V X. _V ) -> ( A e. _I <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I ) )
19	18	biimprd 238	. . . 4 |- ( A e. ( _V X. _V ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I -> A e. _I ) )
20	15, 19	syl5bir 233	. . 3 |- ( A e. ( _V X. _V ) -> ( ( 1st ` A ) = ( 2nd ` A ) -> A e. _I ) )
21	20	imp 445	. 2 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) -> A e. _I )
22	17, 21	impbii 199	1 |- ( A e. _I <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   {copab 4712   _I cid 5023   X. cxp 5112   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  bj-elid2  33086  bj-elid3  33087