Theorem 1stval2 6351

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 4814	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2818	. . . . . 6 |- x e. _V
3		vex 2818	. . . . . 6 |- y e. _V
4	2, 3	op1st 6342	. . . . 5 |- ( 1st ` <. x , y >. ) = x
5	2, 3	op1stb 4601	. . . . 5 |- |^| |^| <. x , y >. = x
6	4, 5	eqtr4i 2258	. . . 4 |- ( 1st ` <. x , y >. ) = |^| |^| <. x , y >.
7		fveq2 5672	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = ( 1st ` <. x , y >. ) )
8		inteq 3954	. . . . 5 |- ( A = <. x , y >. -> |^| A = |^| <. x , y >. )
9	8	inteqd 3956	. . . 4 |- ( A = <. x , y >. -> |^| |^| A = |^| |^| <. x , y >. )
10	6, 7, 9	3eqtr4a 2293	. . 3 |- ( A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
11	10	exlimivv 1948	. 2 |- ( E. x E. y A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
12	1, 11	sylbi 121	1 |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   <.cop 3694   |^|cint 3951   X. cxp 4749   ` cfv 5354   1stc1st 6334

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-1st 6336

This theorem is referenced by:  1stdm  6378