Theorem 2ndval2 6318

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 4788	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2805	. . . . . 6 |- x e. _V
3		vex 2805	. . . . . 6 |- y e. _V
4	2, 3	op2nd 6309	. . . . 5 |- ( 2nd ` <. x , y >. ) = y
5	2, 3	op2ndb 5220	. . . . 5 |- |^| |^| |^| `' { <. x , y >. } = y
6	4, 5	eqtr4i 2255	. . . 4 |- ( 2nd ` <. x , y >. ) = |^| |^| |^| `' { <. x , y >. }
7		fveq2 5639	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = ( 2nd ` <. x , y >. ) )
8		sneq 3680	. . . . . . . 8 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
9	8	cnveqd 4906	. . . . . . 7 |- ( A = <. x , y >. -> `' { A } = `' { <. x , y >. } )
10	9	inteqd 3933	. . . . . 6 |- ( A = <. x , y >. -> |^| `' { A } = |^| `' { <. x , y >. } )
11	10	inteqd 3933	. . . . 5 |- ( A = <. x , y >. -> |^| |^| `' { A } = |^| |^| `' { <. x , y >. } )
12	11	inteqd 3933	. . . 4 |- ( A = <. x , y >. -> |^| |^| |^| `' { A } = |^| |^| |^| `' { <. x , y >. } )
13	6, 7, 12	3eqtr4a 2290	. . 3 |- ( A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
14	13	exlimivv 1945	. 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 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   |^|cint 3928   X. cxp 4723   `'ccnv 4724   ` cfv 5326   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-int 3929  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-iota 5286  df-fun 5328  df-fv 5334  df-2nd 6303

This theorem is referenced by: (None)