Theorem df1st2 6195

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 6133	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5420	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3		dffn5im 5540	. . . . 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 4084	. . . 4 |- ( w e. _V |-> ( 1st ` w ) ) = { <. w , z >. | z = ( 1st ` w ) }
6	4, 5	eqtri 2191	. . 3 |- 1st = { <. w , z >. | z = ( 1st ` w ) }
7	6	reseq1i 4885	. 2 |- ( 1st |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) )
8		resopab 4933	. 2 |- ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) }
9		vex 2733	. . . . 5 |- x e. _V
10		vex 2733	. . . . 5 |- y e. _V
11	9, 10	op1std 6124	. . . 4 |- ( w = <. x , y >. -> ( 1st ` w ) = x )
12	11	eqeq2d 2182	. . 3 |- ( w = <. x , y >. -> ( z = ( 1st ` w ) <-> z = x ) )
13	12	dfoprab3 6167	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) } = { <. <. x , y >. , z >. | z = x }
14	7, 8, 13	3eqtrri 2196	1 |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   <.cop 3584   {copab 4047   |-> cmpt 4048   X. cxp 4607   |` cres 4611   Fn wfn 5191   -onto->wfo 5194   ` cfv 5196   {coprab 5851   1stc1st 6114

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204  df-oprab 5854  df-1st 6116  df-2nd 6117

This theorem is referenced by: (None)