Theorem 2ndval2 6350

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 4812	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2816	. . . . . 6 |- x e. _V
3		vex 2816	. . . . . 6 |- y e. _V
4	2, 3	op2nd 6341	. . . . 5 |- ( 2nd ` <. x , y >. ) = y
5	2, 3	op2ndb 5246	. . . . 5 |- |^| |^| |^| `' { <. x , y >. } = y
6	4, 5	eqtr4i 2256	. . . 4 |- ( 2nd ` <. x , y >. ) = |^| |^| |^| `' { <. x , y >. }
7		fveq2 5670	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = ( 2nd ` <. x , y >. ) )
8		sneq 3700	. . . . . . . 8 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
9	8	cnveqd 4931	. . . . . . 7 |- ( A = <. x , y >. -> `' { A } = `' { <. x , y >. } )
10	9	inteqd 3954	. . . . . 6 |- ( A = <. x , y >. -> |^| `' { A } = |^| `' { <. x , y >. } )
11	10	inteqd 3954	. . . . 5 |- ( A = <. x , y >. -> |^| |^| `' { A } = |^| |^| `' { <. x , y >. } )
12	11	inteqd 3954	. . . 4 |- ( A = <. x , y >. -> |^| |^| |^| `' { A } = |^| |^| |^| `' { <. x , y >. } )
13	6, 7, 12	3eqtr4a 2291	. . 3 |- ( A = <. x , y >. -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
14	13	exlimivv 1946	. 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 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   {csn 3689   <.cop 3692   |^|cint 3949   X. cxp 4747   `'ccnv 4748   ` cfv 5352   2ndc2nd 6333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-2nd 6335

This theorem is referenced by: (None)