Theorem 1stval2 5802

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

Assertion

Ref	Expression
1stval2	|- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )

Proof of Theorem 1stval2

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	op1st 5793	. . . . 5 |- ( 1st ` <. x , y >. ) = x
5	2, 3	op1stb 4227	. . . . 5 |- |^| |^| <. x , y >. = x
6	4, 5	eqtr4i 2104	. . . 4 |- ( 1st ` <. x , y >. ) = |^| |^| <. x , y >.
7		fveq2 5198	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = ( 1st ` <. x , y >. ) )
8		inteq 3639	. . . . 5 |- ( A = <. x , y >. -> |^| A = |^| <. x , y >. )
9	8	inteqd 3641	. . . 4 |- ( A = <. x , y >. -> |^| |^| A = |^| |^| <. x , y >. )
10	6, 7, 9	3eqtr4a 2139	. . 3 |- ( A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
11	10	exlimivv 1817	. 2 |- ( E. x E. y A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
12	1, 11	sylbi 119	1 |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   <.cop 3401   |^|cint 3636   X. cxp 4361   ` cfv 4922   1stc1st 5785

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-1st 5787

This theorem is referenced by:  1stdm  5828