Theorem df1st2 6305

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 6243	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5500	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3		dffn5im 5624	. . . . 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 4141	. . . 4 |- ( w e. _V |-> ( 1st ` w ) ) = { <. w , z >. | z = ( 1st ` w ) }
6	4, 5	eqtri 2226	. . 3 |- 1st = { <. w , z >. | z = ( 1st ` w ) }
7	6	reseq1i 4955	. 2 |- ( 1st |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) )
8		resopab 5003	. 2 |- ( { <. w , z >. | z = ( 1st ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) }
9		vex 2775	. . . . 5 |- x e. _V
10		vex 2775	. . . . 5 |- y e. _V
11	9, 10	op1std 6234	. . . 4 |- ( w = <. x , y >. -> ( 1st ` w ) = x )
12	11	eqeq2d 2217	. . 3 |- ( w = <. x , y >. -> ( z = ( 1st ` w ) <-> z = x ) )
13	12	dfoprab3 6277	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 1st ` w ) ) } = { <. <. x , y >. , z >. | z = x }
14	7, 8, 13	3eqtrri 2231	1 |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   {copab 4104   |-> cmpt 4105   X. cxp 4673   |` cres 4677   Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   {coprab 5945   1stc1st 6224

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5948  df-1st 6226  df-2nd 6227

This theorem is referenced by: (None)