Theorem fsplitOLD

Description: Obsolete proof of fsplit as of 31-Dec-2023. (Contributed by NM, 17-Sep-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	fsplitOLD	|- `' ( 1st |` _I ) = ( x e. _V |-> <. x , x >. )

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2		vex	|- y e. _V
3	1 2	brcnv	|- ( x `' ( 1st |` _I ) y <-> y ( 1st |` _I ) x )
4	1	brresi	|- ( y ( 1st |` _I ) x <-> ( y e. _I /\ y 1st x ) )
5		19.42v	|- ( E. z ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> ( ( 1st ` y ) = x /\ E. z y = <. z , z >. ) )
6		vex	|- z e. _V
7	6 6	op1std	|- ( y = <. z , z >. -> ( 1st ` y ) = z )
8	7	eqeq1d	|- ( y = <. z , z >. -> ( ( 1st ` y ) = x <-> z = x ) )
9	8	pm5.32ri	|- ( ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> ( z = x /\ y = <. z , z >. ) )
10	9	exbii	|- ( E. z ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> E. z ( z = x /\ y = <. z , z >. ) )
11		fo1st	|- 1st : _V -onto-> _V
12		fofn	|- ( 1st : _V -onto-> _V -> 1st Fn _V )
13	11 12	ax-mp	|- 1st Fn _V
14		fnbrfvb	|- ( ( 1st Fn _V /\ y e. _V ) -> ( ( 1st ` y ) = x <-> y 1st x ) )
15	13 2 14	mp2an	|- ( ( 1st ` y ) = x <-> y 1st x )
16		dfid2	|- _I = { <. z , z >. | z = z }
17	16	eleq2i	|- ( y e. _I <-> y e. { <. z , z >. | z = z } )
18		nfe1	|- F/ z E. z ( y = <. z , z >. /\ z = z )
19	18	19.9	|- ( E. z E. z ( y = <. z , z >. /\ z = z ) <-> E. z ( y = <. z , z >. /\ z = z ) )
20		elopab	|- ( y e. { <. z , z >. | z = z } <-> E. z E. z ( y = <. z , z >. /\ z = z ) )
21		equid	|- z = z
22	21	biantru	|- ( y = <. z , z >. <-> ( y = <. z , z >. /\ z = z ) )
23	22	exbii	|- ( E. z y = <. z , z >. <-> E. z ( y = <. z , z >. /\ z = z ) )
24	19 20 23	3bitr4i	|- ( y e. { <. z , z >. | z = z } <-> E. z y = <. z , z >. )
25	17 24	bitr2i	|- ( E. z y = <. z , z >. <-> y e. _I )
26	15 25	anbi12ci	|- ( ( ( 1st ` y ) = x /\ E. z y = <. z , z >. ) <-> ( y e. _I /\ y 1st x ) )
27	5 10 26	3bitr3ri	|- ( ( y e. _I /\ y 1st x ) <-> E. z ( z = x /\ y = <. z , z >. ) )
28		id	|- ( z = x -> z = x )
29	28 28	opeq12d	|- ( z = x -> <. z , z >. = <. x , x >. )
30	29	eqeq2d	|- ( z = x -> ( y = <. z , z >. <-> y = <. x , x >. ) )
31	30	equsexvw	|- ( E. z ( z = x /\ y = <. z , z >. ) <-> y = <. x , x >. )
32	27 31	bitri	|- ( ( y e. _I /\ y 1st x ) <-> y = <. x , x >. )
33	4 32	bitri	|- ( y ( 1st |` _I ) x <-> y = <. x , x >. )
34	3 33	bitri	|- ( x `' ( 1st |` _I ) y <-> y = <. x , x >. )
35	34	opabbii	|- { <. x , y >. | x `' ( 1st |` _I ) y } = { <. x , y >. | y = <. x , x >. }
36		relcnv	|- Rel `' ( 1st |` _I )
37		dfrel4v	|- ( Rel `' ( 1st |` _I ) <-> `' ( 1st |` _I ) = { <. x , y >. | x `' ( 1st |` _I ) y } )
38	36 37	mpbi	|- `' ( 1st |` _I ) = { <. x , y >. | x `' ( 1st |` _I ) y }
39		mptv	|- ( x e. _V |-> <. x , x >. ) = { <. x , y >. | y = <. x , x >. }
40	35 38 39	3eqtr4i	|- `' ( 1st |` _I ) = ( x e. _V |-> <. x , x >. )