Theorem 2ndval2 6147

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 4682	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2738	. . . . . 6 |- x e. _V
3		vex 2738	. . . . . 6 |- y e. _V
4	2, 3	op2nd 6138	. . . . 5 |- ( 2nd ` <. x , y >. ) = y
5	2, 3	op2ndb 5104	. . . . 5 |- |^| |^| |^| `' { <. x , y >. } = y
6	4, 5	eqtr4i 2199	. . . 4 |- ( 2nd ` <. x , y >. ) = |^| |^| |^| `' { <. x , y >. }
7		fveq2 5507	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = ( 2nd ` <. x , y >. ) )
8		sneq 3600	. . . . . . . 8 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
9	8	cnveqd 4796	. . . . . . 7 |- ( A = <. x , y >. -> `' { A } = `' { <. x , y >. } )
10	9	inteqd 3845	. . . . . 6 |- ( A = <. x , y >. -> |^| `' { A } = |^| `' { <. x , y >. } )
11	10	inteqd 3845	. . . . 5 |- ( A = <. x , y >. -> |^| |^| `' { A } = |^| |^| `' { <. x , y >. } )
12	11	inteqd 3845	. . . 4 |- ( A = <. x , y >. -> |^| |^| |^| `' { A } = |^| |^| |^| `' { <. x , y >. } )
13	6, 7, 12	3eqtr4a 2234	. . 3 |- ( A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
14	13	exlimivv 1894	. 2 |- ( E. x E. y A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
15	1, 14	sylbi 121	1 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) = |^| |^| |^| `' { A } )

Syntax hints:   -> wi 4   = wceq 1353   E.wex 1490   e. wcel 2146   _Vcvv 2735   {csn 3589   <.cop 3592   |^|cint 3840   X. cxp 4618   `'ccnv 4619   ` cfv 5208   2ndc2nd 6130

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fv 5216  df-2nd 6132

This theorem is referenced by: (None)