Theorem op1stg 6000

Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Assertion

Ref	Expression
op1stg	|- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )

Proof of Theorem op1stg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3669	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5377	. . 3 |- ( x = A -> ( 1st ` <. x , y >. ) = ( 1st ` <. A , y >. ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2127	. 2 |- ( x = A -> ( ( 1st ` <. x , y >. ) = x <-> ( 1st ` <. A , y >. ) = A ) )
5		opeq2 3670	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
6	5	fveq2d 5377	. . 3 |- ( y = B -> ( 1st ` <. A , y >. ) = ( 1st ` <. A , B >. ) )
7	6	eqeq1d 2121	. 2 |- ( y = B -> ( ( 1st ` <. A , y >. ) = A <-> ( 1st ` <. A , B >. ) = A ) )
8		vex 2658	. . 3 |- x e. _V
9		vex 2658	. . 3 |- y e. _V
10	8, 9	op1st 5996	. 2 |- ( 1st ` <. x , y >. ) = x
11	4, 7, 10	vtocl2g 2719	1 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   <.cop 3494   ` cfv 5079   1stc1st 5988

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fv 5087  df-1st 5990

This theorem is referenced by:  ot1stg  6002  ot2ndg  6003  1stconst  6070  algrflemg  6079  mpoxopn0yelv  6088  mpoxopoveq  6089  xpmapenlem  6694  1stinl  6909  1stinr  6911  mulpipq  7122  frecuzrdgg  10076  qredeu  11618  qnumdenbi  11709  upxp  12277  uptx  12279  txmetcnp  12501