Theorem df1st2

Description: An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Assertion	df1st2	|- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )

Proof

Step	Hyp	Ref	Expression
1		fo1st	|- 1st : _V -onto-> _V
2		fofn	|- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1 2	ax-mp	|- 1st Fn _V
4		dffn5	|- ( 1st Fn _V <-> 1st = ( w e. _V |-> ( 1st ` w ) ) )
5	3 4	mpbi	|- 1st = ( w e. _V |-> ( 1st ` w ) )
6		mptv	|- ( w e. _V |-> ( 1st ` w ) ) = { <. w , z >. | z = ( 1st ` w ) }
7	5 6	eqtri	|- 1st = { <. w , z >. | z = ( 1st ` w ) }
8	7	reseq1i	|- ( 1st |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) )
9		resopab	|- ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) }
10		vex	|- x e. _V
11		vex	|- y e. _V
12	10 11	op1std	|- ( w = <. x , y >. -> ( 1st ` w ) = x )
13	12	eqeq2d	|- ( w = <. x , y >. -> ( z = ( 1st ` w ) <-> z = x ) )
14	13	dfoprab3	|- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) } = { <. <. x , y >. , z >. | z = x }
15	8 9 14	3eqtrri	|- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )