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Theorem cnmpt1st 20625
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 6771 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5755 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 ssv 3427 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5650 . . . . 5  |-  ( ( 1st  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 676 . . . 4  |-  ( 1st  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5870 . . . 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 211 . . 3  |-  ( 1st  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5839 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
109mpteq2ia 4449 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 1st  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 1st `  z
) )
11 vex 3025 . . . . 5  |-  x  e. 
_V
12 vex 3025 . . . . 5  |-  y  e. 
_V
1311, 12op1std 6761 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
1413mpt2mpt 6346 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 1st `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  x )
158, 10, 143eqtri 2454 . 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 20566 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  J ) )
1916, 17, 18syl2anc 665 . 2  |-  ( ph  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  J ) )
2015, 19syl5eqelr 2511 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 1437    e. wcel 1872   _Vcvv 3022    C_ wss 3379    |-> cmpt 4425    X. cxp 4794    |` cres 4798    Fn wfn 5539   -onto->wfo 5542   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   1stc1st 6749  TopOnctopon 19860    Cn ccn 20182    tX ctx 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fo 5550  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-map 7429  df-topgen 15285  df-top 19863  df-bases 19864  df-topon 19865  df-cn 20185  df-tx 20519
This theorem is referenced by:  cnmptcom  20635  xkofvcn  20641  cnmptk2  20643  txhmeo  20760  txswaphmeo  20762  ptunhmeo  20765  xkohmeo  20772  tgpsubcn  21047  istgp2  21048  oppgtmd  21054  prdstmdd  21080  dvrcn  21140  divcn  21842  cnrehmeo  21923  htpycom  21949  htpyid  21950  htpyco1  21951  htpycc  21953  reparphti  21970  pcocn  21990  pcohtpylem  21992  pcopt  21995  pcopt2  21996  pcoass  21997  pcorevlem  21999  cxpcn  23627  vmcn  26277  dipcn  26301  mndpluscn  28684  cvxscon  29918  cvmlift2lem12  29989
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