Theorem op1stg 6095

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 3741	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5471	. . 3 |- ( x = A -> ( 1st ` <. x , y >. ) = ( 1st ` <. A , y >. ) )
3		id 19	. . 3 |- ( x = A -> x = A )
4	2, 3	eqeq12d 2172	. 2 |- ( x = A -> ( ( 1st ` <. x , y >. ) = x <-> ( 1st ` <. A , y >. ) = A ) )
5		opeq2 3742	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
6	5	fveq2d 5471	. . 3 |- ( y = B -> ( 1st ` <. A , y >. ) = ( 1st ` <. A , B >. ) )
7	6	eqeq1d 2166	. 2 |- ( y = B -> ( ( 1st ` <. A , y >. ) = A <-> ( 1st ` <. A , B >. ) = A ) )
8		vex 2715	. . 3 |- x e. _V
9		vex 2715	. . 3 |- y e. _V
10	8, 9	op1st 6091	. 2 |- ( 1st ` <. x , y >. ) = x
11	4, 7, 10	vtocl2g 2776	1 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   <.cop 3563   ` cfv 5169   1stc1st 6083

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fv 5177  df-1st 6085

This theorem is referenced by:  ot1stg  6097  ot2ndg  6098  1stconst  6165  algrflemg  6174  mpoxopn0yelv  6183  mpoxopoveq  6184  xpmapenlem  6791  1stinl  7012  1stinr  7014  mulpipq  7286  suplocexprlemlub  7638  aprcl  8515  frecuzrdgg  10308  qredeu  11965  qnumdenbi  12057  upxp  12643  uptx  12645  txmetcnp  12889