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Theorem cnmpt1st 17362
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 6139 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5453 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  1st  Fn  _V
4 ssv 3198 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5357 . . . . 5  |-  ( ( 1st  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 653 . . . 4  |-  ( 1st  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5568 . . . 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 199 . . 3  |-  ( 1st  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5542 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
109mpteq2ia 4102 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 1st `  z
) )
11 vex 2791 . . . . 5  |-  x  e. 
_V
12 vex 2791 . . . . 5  |-  y  e. 
_V
1311, 12op1std 6130 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
1413mpt2mpt 5939 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 1st `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  x )
158, 10, 143eqtri 2307 . 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 17303 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  J ) )
1916, 17, 18syl2anc 642 . 2  |-  ( ph  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  J ) )
2015, 19syl5eqelr 2368 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    e. cmpt 4077    X. cxp 4687    |` cres 4691    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255
This theorem is referenced by:  cnmptcom  17372  xkofvcn  17378  cnmptk2  17380  txhmeo  17494  txswaphmeo  17496  ptunhmeo  17499  xkohmeo  17506  tgpsubcn  17773  istgp2  17774  oppgtmd  17780  prdstmdd  17806  dvrcn  17866  divcn  18372  cnrehmeo  18451  htpycom  18474  htpyid  18475  htpyco1  18476  htpycc  18478  reparphti  18495  pcocn  18515  pcohtpylem  18517  pcopt  18520  pcopt2  18521  pcoass  18522  pcorevlem  18524  cxpcn  20085  vmcn  21272  dipcn  21296  mndpluscn  23299  cvxscon  23774  cvmlift2lem12  23845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257
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