Theorem 1stval2 6134

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 4673	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2733	. . . . . 6 |- x e. _V
3		vex 2733	. . . . . 6 |- y e. _V
4	2, 3	op1st 6125	. . . . 5 |- ( 1st ` <. x , y >. ) = x
5	2, 3	op1stb 4463	. . . . 5 |- |^| |^| <. x , y >. = x
6	4, 5	eqtr4i 2194	. . . 4 |- ( 1st ` <. x , y >. ) = |^| |^| <. x , y >.
7		fveq2 5496	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = ( 1st ` <. x , y >. ) )
8		inteq 3834	. . . . 5 |- ( A = <. x , y >. -> |^| A = |^| <. x , y >. )
9	8	inteqd 3836	. . . 4 |- ( A = <. x , y >. -> |^| |^| A = |^| |^| <. x , y >. )
10	6, 7, 9	3eqtr4a 2229	. . 3 |- ( A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
11	10	exlimivv 1889	. 2 |- ( E. x E. y A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
12	1, 11	sylbi 120	1 |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )

Syntax hints:   -> wi 4   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586   |^|cint 3831   X. cxp 4609   ` cfv 5198   1stc1st 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119

This theorem is referenced by:  1stdm  6161