Theorem op1stg 5797

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 3570	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5202	. . 3 |- ( x = A -> ( 1st ` <. x , y >. ) = ( 1st ` <. A , y >. ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2095	. 2 |- ( x = A -> ( ( 1st ` <. x , y >. ) = x <-> ( 1st ` <. A , y >. ) = A ) )
5		opeq2 3571	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
6	5	fveq2d 5202	. . 3 |- ( y = B -> ( 1st ` <. A , y >. ) = ( 1st ` <. A , B >. ) )
7	6	eqeq1d 2089	. 2 |- ( y = B -> ( ( 1st ` <. A , y >. ) = A <-> ( 1st ` <. A , B >. ) = A ) )
8		vex 2604	. . 3 |- x e. _V
9		vex 2604	. . 3 |- y e. _V
10	8, 9	op1st 5793	. 2 |- ( 1st ` <. x , y >. ) = x
11	4, 7, 10	vtocl2g 2662	1 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   <.cop 3401   ` cfv 4922   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-1st 5787

This theorem is referenced by:  ot1stg  5799  ot2ndg  5800  1stconst  5862  algrflemg  5871  mpt2xopn0yelv  5877  mpt2xopoveq  5878  mulpipq  6562  qredeu  10479