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Theorem op1stg 6711
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 4131 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5778 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2404 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 4132 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5778 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2384 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 3037 . . 3  |-  x  e. 
_V
9 vex 3037 . . 3  |-  y  e. 
_V
108, 9op1st 6707 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 3096 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   <.cop 3950   ` cfv 5496   1stc1st 6697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fv 5504  df-1st 6699
This theorem is referenced by:  ot1stg  6713  ot2ndg  6714  1stconst  6787  mpt2sn  6790  curry2  6794  mpt2xopn0yelv  6859  mpt2xopoveq  6865  xpmapenlem  7603  fpwwe  8935  addpipq  9226  mulpipq  9229  ordpipq  9231  swrdval  12553  ruclem1  13966  qnumdenbi  14279  oppccofval  15122  funcf2  15274  cofuval2  15293  resfval2  15299  resf1st  15300  isnat  15353  fucco  15368  homadm  15436  setcco  15479  estrcco  15516  xpcco  15569  xpchom2  15572  xpcco2  15573  evlf2  15604  curfval  15609  curf1cl  15614  uncf1  15622  uncf2  15623  diag11  15629  diag12  15630  diag2  15631  hof2fval  15641  yonedalem21  15659  yonedalem22  15664  mvmulfval  19129  imasdsf1olem  20961  ovolicc1  22012  ioombl1lem3  22055  ioombl1lem4  22056  brcgr  24324  nbgraop  24544  rngoablo2  25541  vcoprne  25589  fgreu  27659  fvtransport  29835  etransclem44  32227  gordopval  32708  rngccoALTV  32996  ringccoALTV  33059  bj-elid3  34949  bj-inftyexpiinv  34958  bj-finsumval0  35010  dvhopvadd  37233  dvhopvsca  37242  dvhopaddN  37254  dvhopspN  37255
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