Theorem op1stg 6377

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 3902	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5697	. . 3 |- ( x = A -> ( 1st ` <. x , y >. ) = ( 1st ` <. A , y >. ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2253	. 2 |- ( x = A -> ( ( 1st ` <. x , y >. ) = x <-> ( 1st ` <. A , y >. ) = A ) )
5		opeq2 3903	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
6	5	fveq2d 5697	. . 3 |- ( y = B -> ( 1st ` <. A , y >. ) = ( 1st ` <. A , B >. ) )
7	6	eqeq1d 2247	. 2 |- ( y = B -> ( ( 1st ` <. A , y >. ) = A <-> ( 1st ` <. A , B >. ) = A ) )
8		vex 2824	. . 3 |- x e. _V
9		vex 2824	. . 3 |- y e. _V
10	8, 9	op1st 6373	. 2 |- ( 1st ` <. x , y >. ) = x
11	4, 7, 10	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3711   ` cfv 5375   1stc1st 6365

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-1st 6367

This theorem is referenced by:  ot1stg  6379  ot2ndg  6380  1stconst  6450  algrflemg  6459  mpoxopn0yelv  6503  mpoxopoveq  6504  xpmapenlem  7142  1stinl  7407  1stinr  7409  mulpipq  7732  suplocexprlemlub  8084  aprcl  8967  frecuzrdgg  10834  swrdval  11401  qredeu  12856  qnumdenbi  12951  upxp  15299  uptx  15301  txmetcnp  15545  opvtxfv  16180