Theorem 1stval2 6301

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 4781	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 2802	. . . . . 6 |- x e. _V
3		vex 2802	. . . . . 6 |- y e. _V
4	2, 3	op1st 6292	. . . . 5 |- ( 1st ` <. x , y >. ) = x
5	2, 3	op1stb 4569	. . . . 5 |- |^| |^| <. x , y >. = x
6	4, 5	eqtr4i 2253	. . . 4 |- ( 1st ` <. x , y >. ) = |^| |^| <. x , y >.
7		fveq2 5627	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = ( 1st ` <. x , y >. ) )
8		inteq 3926	. . . . 5 |- ( A = <. x , y >. -> |^| A = |^| <. x , y >. )
9	8	inteqd 3928	. . . 4 |- ( A = <. x , y >. -> |^| |^| A = |^| |^| <. x , y >. )
10	6, 7, 9	3eqtr4a 2288	. . 3 |- ( A = <. x , y >. -> ( 1st ` A ) = |^| |^| A )
11	10	exlimivv 1943	. 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 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   <.cop 3669   |^|cint 3923   X. cxp 4717   ` cfv 5318   1stc1st 6284

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-1st 6286

This theorem is referenced by:  1stdm  6328