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Theorem cnmpt1st 19897
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 6794 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5788 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 ssv 3517 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5685 . . . . 5  |-  ( ( 1st  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 672 . . . 4  |-  ( 1st  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5904 . . . 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 208 . . 3  |-  ( 1st  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5871 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
109mpteq2ia 4522 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 1st `  z
) )
11 vex 3109 . . . . 5  |-  x  e. 
_V
12 vex 3109 . . . . 5  |-  y  e. 
_V
1311, 12op1std 6784 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
1413mpt2mpt 6369 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 1st `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  x )
158, 10, 143eqtri 2493 . 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 19838 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  J ) )
1916, 17, 18syl2anc 661 . 2  |-  ( ph  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  J ) )
2015, 19syl5eqelr 2553 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J  tX  K
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469    |-> cmpt 4498    X. cxp 4990    |` cres 4994    Fn wfn 5574   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772  TopOnctopon 19155    Cn ccn 19484    tX ctx 19789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cn 19487  df-tx 19791
This theorem is referenced by:  cnmptcom  19907  xkofvcn  19913  cnmptk2  19915  txhmeo  20032  txswaphmeo  20034  ptunhmeo  20037  xkohmeo  20044  tgpsubcn  20317  istgp2  20318  oppgtmd  20324  prdstmdd  20350  dvrcn  20414  divcn  21100  cnrehmeo  21181  htpycom  21204  htpyid  21205  htpyco1  21206  htpycc  21208  reparphti  21225  pcocn  21245  pcohtpylem  21247  pcopt  21250  pcopt2  21251  pcoass  21252  pcorevlem  21254  cxpcn  22840  vmcn  25271  dipcn  25295  mndpluscn  27530  cvxscon  28314  cvmlift2lem12  28385
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