Theorem 2ndval2 5803

Description: Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Assertion

Ref	Expression
2ndval2	|- ( A e. ( _V X. _V ) -> ( 2nd ` A ) = |^| |^| |^| `' { A } )

Proof of Theorem 2ndval2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 4420	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2604	. . . . . 6 |- x e. _V
3		vex 2604	. . . . . 6 |- y e. _V
4	2, 3	op2nd 5794	. . . . 5 |- ( 2nd ` <. x , y >. ) = y
5	2, 3	op2ndb 4824	. . . . 5 |- |^| |^| |^| `' { <. x , y >. } = y
6	4, 5	eqtr4i 2104	. . . 4 |- ( 2nd ` <. x , y >. ) = |^| |^| |^| `' { <. x , y >. }
7		fveq2 5198	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = ( 2nd ` <. x , y >. ) )
8		sneq 3409	. . . . . . . 8 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
9	8	cnveqd 4529	. . . . . . 7 |- ( A = <. x , y >. -> `' { A } = `' { <. x , y >. } )
10	9	inteqd 3641	. . . . . 6 |- ( A = <. x , y >. -> |^| `' { A } = |^| `' { <. x , y >. } )
11	10	inteqd 3641	. . . . 5 |- ( A = <. x , y >. -> |^| |^| `' { A } = |^| |^| `' { <. x , y >. } )
12	11	inteqd 3641	. . . 4 |- ( A = <. x , y >. -> |^| |^| |^| `' { A } = |^| |^| |^| `' { <. x , y >. } )
13	6, 7, 12	3eqtr4a 2139	. . 3 |- ( A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
14	13	exlimivv 1817	. 2 |- ( E. x E. y A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
15	1, 14	sylbi 119	1 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) = |^| |^| |^| `' { A } )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   {csn 3398   <.cop 3401   |^|cint 3636   X. cxp 4361   `'ccnv 4362   ` cfv 4922   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-2nd 5788

This theorem is referenced by: (None)