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Theorem cnmpt1st 19216
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 6591 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5617 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 ssv 3371 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5519 . . . . 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 5732 . . . 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 5699 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
109mpteq2ia 4369 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 1st `  z
) )
11 vex 2970 . . . . 5  |-  x  e. 
_V
12 vex 2970 . . . . 5  |-  y  e. 
_V
1311, 12op1std 6582 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
1413mpt2mpt 6177 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 1st `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  x )
158, 10, 143eqtri 2462 . 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 19157 . . 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 2523 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 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323    e. cmpt 4345    X. cxp 4833    |` cres 4837    Fn wfn 5408   -onto->wfo 5411   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570  TopOnctopon 18474    Cn ccn 18803    tX ctx 19108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-map 7208  df-topgen 14374  df-top 18478  df-bases 18480  df-topon 18481  df-cn 18806  df-tx 19110
This theorem is referenced by:  cnmptcom  19226  xkofvcn  19232  cnmptk2  19234  txhmeo  19351  txswaphmeo  19353  ptunhmeo  19356  xkohmeo  19363  tgpsubcn  19636  istgp2  19637  oppgtmd  19643  prdstmdd  19669  dvrcn  19733  divcn  20419  cnrehmeo  20500  htpycom  20523  htpyid  20524  htpyco1  20525  htpycc  20527  reparphti  20544  pcocn  20564  pcohtpylem  20566  pcopt  20569  pcopt2  20570  pcoass  20571  pcorevlem  20573  cxpcn  22158  vmcn  24045  dipcn  24069  mndpluscn  26308  cvxscon  27084  cvmlift2lem12  27155
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