Theorem 1stval2 6152

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 4687	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2740	. . . . . 6 |- x e. _V
3		vex 2740	. . . . . 6 |- y e. _V
4	2, 3	op1st 6143	. . . . 5 |- ( 1st ` <. x , y >. ) = x
5	2, 3	op1stb 4477	. . . . 5 |- |^| |^| <. x , y >. = x
6	4, 5	eqtr4i 2201	. . . 4 |- ( 1st ` <. x , y >. ) = |^| |^| <. x , y >.
7		fveq2 5513	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = ( 1st ` <. x , y >. ) )
8		inteq 3847	. . . . 5 |- ( A = <. x , y >. -> |^| A = |^| <. x , y >. )
9	8	inteqd 3849	. . . 4 |- ( A = <. x , y >. -> |^| |^| A = |^| |^| <. x , y >. )
10	6, 7, 9	3eqtr4a 2236	. . 3 |- ( A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
11	10	exlimivv 1896	. 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 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   <.cop 3595   |^|cint 3844   X. cxp 4623   ` cfv 5214   1stc1st 6135

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5176  df-fun 5216  df-fv 5222  df-1st 6137

This theorem is referenced by:  1stdm  6179