Theorem df1st2 6222

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

Assertion

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

Distinct variable group:   x, y, z

Proof of Theorem df1st2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 6160	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5442	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3		dffn5im 5563	. . . . 5 |- ( 1st Fn _V -> 1st = ( w e. _V |-> ( 1st ` w ) ) )
4	1, 2, 3	mp2b 8	. . . 4 |- 1st = ( w e. _V |-> ( 1st ` w ) )
5		mptv 4102	. . . 4 |- ( w e. _V |-> ( 1st ` w ) ) = { <. w , z >. | z = ( 1st ` w ) }
6	4, 5	eqtri 2198	. . 3 |- 1st = { <. w , z >. | z = ( 1st ` w ) }
7	6	reseq1i 4905	. 2 |- ( 1st |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) )
8		resopab 4953	. 2 |- ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) }
9		vex 2742	. . . . 5 |- x e. _V
10		vex 2742	. . . . 5 |- y e. _V
11	9, 10	op1std 6151	. . . 4 |- ( w = <. x , y >. -> ( 1st ` w ) = x )
12	11	eqeq2d 2189	. . 3 |- ( w = <. x , y >. -> ( z = ( 1st ` w ) <-> z = x ) )
13	12	dfoprab3 6194	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) } = { <. <. x , y >. , z >. | z = x }
14	7, 8, 13	3eqtrri 2203	1 |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   <.cop 3597   {copab 4065   |-> cmpt 4066   X. cxp 4626   |` cres 4630   Fn wfn 5213   -onto->wfo 5216   ` cfv 5218   {coprab 5878   1stc1st 6141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226  df-oprab 5881  df-1st 6143  df-2nd 6144

This theorem is referenced by: (None)