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Theorem cnmpt1st 17621
Description: The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmpt21.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
Assertion
Ref Expression
cnmpt1st  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J  tX  K
)  Cn  J ) )
Distinct variable groups:    x, y, ph    x, X, y    x, Y, y
Allowed substitution hints:    J( x, y)    K( x, y)

Proof of Theorem cnmpt1st
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fo1st 6305 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5595 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  1st  Fn  _V
4 ssv 3311 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5498 . . . . 5  |-  ( ( 1st  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 654 . . . 4  |-  ( 1st  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5711 . . . 4  |-  ( ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
)  <->  ( 1st  |`  ( X  X.  Y ) )  =  ( z  e.  ( X  X.  Y
)  |->  ( ( 1st  |`  ( X  X.  Y
) ) `  z
) ) )
86, 7mpbi 200 . . 3  |-  ( 1st  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5685 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
109mpteq2ia 4232 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 1st `  z
) )
11 vex 2902 . . . . 5  |-  x  e. 
_V
12 vex 2902 . . . . 5  |-  y  e. 
_V
1311, 12op1std 6296 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
1413mpt2mpt 6104 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 1st `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  x )
158, 10, 143eqtri 2411 . 2  |-  ( 1st  |`  ( X  X.  Y
) )  =  ( x  e.  X , 
y  e.  Y  |->  x )
16 cnmpt21.j . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
17 cnmpt21.k . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
18 tx1cn 17562 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  J ) )
1916, 17, 18syl2anc 643 . 2  |-  ( ph  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  J ) )
2015, 19syl5eqelr 2472 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J  tX  K
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263    e. cmpt 4207    X. cxp 4816    |` cres 4820    Fn wfn 5389   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286  TopOnctopon 16882    Cn ccn 17210    tX ctx 17513
This theorem is referenced by:  cnmptcom  17631  xkofvcn  17637  cnmptk2  17639  txhmeo  17756  txswaphmeo  17758  ptunhmeo  17761  xkohmeo  17768  tgpsubcn  18041  istgp2  18042  oppgtmd  18048  prdstmdd  18074  dvrcn  18134  divcn  18769  cnrehmeo  18849  htpycom  18872  htpyid  18873  htpyco1  18874  htpycc  18876  reparphti  18893  pcocn  18913  pcohtpylem  18915  pcopt  18918  pcopt2  18919  pcoass  18920  pcorevlem  18922  cxpcn  20496  vmcn  22043  dipcn  22067  mndpluscn  24116  cvxscon  24709  cvmlift2lem12  24780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956  df-topgen 13594  df-top 16886  df-bases 16888  df-topon 16889  df-cn 17213  df-tx 17515
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