Theorem df1st2 6187

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 6125	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5412	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3		dffn5im 5532	. . . . 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 4079	. . . 4 |- ( w e. _V |-> ( 1st ` w ) ) = { <. w , z >. | z = ( 1st ` w ) }
6	4, 5	eqtri 2186	. . 3 |- 1st = { <. w , z >. | z = ( 1st ` w ) }
7	6	reseq1i 4880	. 2 |- ( 1st |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) )
8		resopab 4928	. 2 |- ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) }
9		vex 2729	. . . . 5 |- x e. _V
10		vex 2729	. . . . 5 |- y e. _V
11	9, 10	op1std 6116	. . . 4 |- ( w = <. x , y >. -> ( 1st ` w ) = x )
12	11	eqeq2d 2177	. . 3 |- ( w = <. x , y >. -> ( z = ( 1st ` w ) <-> z = x ) )
13	12	dfoprab3 6159	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) } = { <. <. x , y >. , z >. | z = x }
14	7, 8, 13	3eqtrri 2191	1 |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   <.cop 3579   {copab 4042   |-> cmpt 4043   X. cxp 4602   |` cres 4606   Fn wfn 5183   -onto->wfo 5186   ` cfv 5188   {coprab 5843   1stc1st 6106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-oprab 5846  df-1st 6108  df-2nd 6109

This theorem is referenced by: (None)