Theorem df2nd2 6329

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 6267	. . . . 5 |- 2nd : _V -onto-> _V
2		fofn 5522	. . . . 5 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3		dffn5im 5647	. . . . 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 4157	. . . 4 |- ( w e. _V |-> ( 2nd ` w ) ) = { <. w , z >. | z = ( 2nd ` w ) }
6	4, 5	eqtri 2228	. . 3 |- 2nd = { <. w , z >. | z = ( 2nd ` w ) }
7	6	reseq1i 4974	. 2 |- ( 2nd |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) )
8		resopab 5022	. 2 |- ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) }
9		vex 2779	. . . . 5 |- x e. _V
10		vex 2779	. . . . 5 |- y e. _V
11	9, 10	op2ndd 6258	. . . 4 |- ( w = <. x , y >. -> ( 2nd ` w ) = y )
12	11	eqeq2d 2219	. . 3 |- ( w = <. x , y >. -> ( z = ( 2nd ` w ) <-> z = y ) )
13	12	dfoprab3 6300	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) } = { <. <. x , y >. , z >. | z = y }
14	7, 8, 13	3eqtrri 2233	1 |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   <.cop 3646   {copab 4120   |-> cmpt 4121   X. cxp 4691   |` cres 4695   Fn wfn 5285   -onto->wfo 5288   ` cfv 5290   {coprab 5968   2ndc2nd 6248

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-oprab 5971  df-1st 6249  df-2nd 6250

This theorem is referenced by: (None)