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Theorem op1stg 6797
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.
StepHypRef Expression
1 opeq1 4202 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5860 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2465 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 4203 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5860 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2445 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 3098 . . 3  |-  x  e. 
_V
9 vex 3098 . . 3  |-  y  e. 
_V
108, 9op1st 6793 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 3157 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   <.cop 4020   ` cfv 5578   1stc1st 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fv 5586  df-1st 6785
This theorem is referenced by:  ot1stg  6799  ot2ndg  6800  1stconst  6873  mpt2sn  6876  curry2  6880  mpt2xopn0yelv  6943  mpt2xopoveq  6949  xpmapenlem  7686  fpwwe  9027  addpipq  9318  mulpipq  9321  ordpipq  9323  swrdval  12623  ruclem1  13841  qnumdenbi  14154  oppccofval  14988  funcf2  15111  cofuval2  15130  resfval2  15136  resf1st  15137  isnat  15190  fucco  15205  homadm  15241  setcco  15284  xpcco  15326  xpchom2  15329  xpcco2  15330  evlf2  15361  curfval  15366  curf1cl  15371  uncf1  15379  uncf2  15380  diag11  15386  diag12  15387  diag2  15388  hof2fval  15398  yonedalem21  15416  yonedalem22  15421  mvmulfval  18917  imasdsf1olem  20749  ovolicc1  21800  ioombl1lem3  21843  ioombl1lem4  21844  brcgr  24075  nbgraop  24295  rngoablo2  25296  vcoprne  25344  fgreu  27385  fvtransport  29657  gordopval  32228  estrcco  32485  rngccoOLD  32536  ringccoOLD  32596  bj-elid3  34341  bj-inftyexpiinv  34351  bj-finsumval0  34403  dvhopvadd  36560  dvhopvsca  36569  dvhopaddN  36581  dvhopspN  36582
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