Theorem df2nd2 6384

Description: An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
df2nd2	|- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Distinct variable group:   x, y, z

Proof of Theorem df2nd2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 6320	. . . . 5 |- 2nd : _V -onto-> _V
2		fofn 5561	. . . . 5 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3		dffn5im 5691	. . . . 5 |- ( 2nd Fn _V -> 2nd = ( w e. _V |-> ( 2nd ` w ) ) )
4	1, 2, 3	mp2b 8	. . . 4 |- 2nd = ( w e. _V |-> ( 2nd ` w ) )
5		mptv 4186	. . . 4 |- ( w e. _V |-> ( 2nd ` w ) ) = { <. w , z >. | z = ( 2nd ` w ) }
6	4, 5	eqtri 2252	. . 3 |- 2nd = { <. w , z >. | z = ( 2nd ` w ) }
7	6	reseq1i 5009	. 2 |- ( 2nd |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) )
8		resopab 5057	. 2 |- ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) }
9		vex 2805	. . . . 5 |- x e. _V
10		vex 2805	. . . . 5 |- y e. _V
11	9, 10	op2ndd 6311	. . . 4 |- ( w = <. x , y >. -> ( 2nd ` w ) = y )
12	11	eqeq2d 2243	. . 3 |- ( w = <. x , y >. -> ( z = ( 2nd ` w ) <-> z = y ) )
13	12	dfoprab3 6353	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) } = { <. <. x , y >. , z >. | z = y }
14	7, 8, 13	3eqtrri 2257	1 |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   {copab 4149   |-> cmpt 4150   X. cxp 4723   |` cres 4727   Fn wfn 5321   -onto->wfo 5324   ` cfv 5326   {coprab 6018   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-oprab 6021  df-1st 6302  df-2nd 6303

This theorem is referenced by: (None)