Theorem 2ndval2 6255

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 4745	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2776	. . . . . 6 |- x e. _V
3		vex 2776	. . . . . 6 |- y e. _V
4	2, 3	op2nd 6246	. . . . 5 |- ( 2nd ` <. x , y >. ) = y
5	2, 3	op2ndb 5175	. . . . 5 |- |^| |^| |^| `' { <. x , y >. } = y
6	4, 5	eqtr4i 2230	. . . 4 |- ( 2nd ` <. x , y >. ) = |^| |^| |^| `' { <. x , y >. }
7		fveq2 5589	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = ( 2nd ` <. x , y >. ) )
8		sneq 3649	. . . . . . . 8 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
9	8	cnveqd 4862	. . . . . . 7 |- ( A = <. x , y >. -> `' { A } = `' { <. x , y >. } )
10	9	inteqd 3896	. . . . . 6 |- ( A = <. x , y >. -> |^| `' { A } = |^| `' { <. x , y >. } )
11	10	inteqd 3896	. . . . 5 |- ( A = <. x , y >. -> |^| |^| `' { A } = |^| |^| `' { <. x , y >. } )
12	11	inteqd 3896	. . . 4 |- ( A = <. x , y >. -> |^| |^| |^| `' { A } = |^| |^| |^| `' { <. x , y >. } )
13	6, 7, 12	3eqtr4a 2265	. . 3 |- ( A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
14	13	exlimivv 1921	. 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 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   |^|cint 3891   X. cxp 4681   `'ccnv 4682   ` cfv 5280   2ndc2nd 6238

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-2nd 6240

This theorem is referenced by: (None)