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Theorem cnmpt2nd 20615
Description: The projection onto the second 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
cnmpt2nd  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J  tX  K
)  Cn  K ) )
Distinct variable groups:    x, y, ph    x, X, y    x, Y, y
Allowed substitution hints:    J( x, y)    K( x, y)

Proof of Theorem cnmpt2nd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fo2nd 6828 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5812 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  2nd  Fn  _V
4 ssv 3490 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5707 . . . . 5  |-  ( ( 2nd  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 676 . . . 4  |-  ( 2nd  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5926 . . . 4  |-  ( ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
)  <->  ( 2nd  |`  ( X  X.  Y ) )  =  ( z  e.  ( X  X.  Y
)  |->  ( ( 2nd  |`  ( X  X.  Y
) ) `  z
) ) )
86, 7mpbi 211 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5895 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
109mpteq2ia 4508 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 2nd `  z
) )
11 vex 3090 . . . . 5  |-  x  e. 
_V
12 vex 3090 . . . . 5  |-  y  e. 
_V
1311, 12op2ndd 6818 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
1413mpt2mpt 6402 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 2nd `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  y )
158, 10, 143eqtri 2462 . 2  |-  ( 2nd  |`  ( X  X.  Y
) )  =  ( x  e.  X , 
y  e.  Y  |->  y )
16 cnmpt21.j . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
17 cnmpt21.k . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
18 tx2cn 20556 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  K ) )
1916, 17, 18syl2anc 665 . 2  |-  ( ph  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  K ) )
2015, 19syl5eqelr 2522 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J  tX  K
)  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442    |-> cmpt 4484    X. cxp 4852    |` cres 4856    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   2ndc2nd 6806  TopOnctopon 19849    Cn ccn 20171    tX ctx 20506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-topgen 15301  df-top 19852  df-bases 19853  df-topon 19854  df-cn 20174  df-tx 20508
This theorem is referenced by:  cnmptcom  20624  xkofvcn  20630  cnmptk2  20632  txhmeo  20749  txswaphmeo  20751  ptunhmeo  20754  xkohmeo  20761  tgpsubcn  21036  istgp2  21037  oppgtmd  21043  prdstmdd  21069  dvrcn  21129  divcn  21796  cnrehmeo  21877  htpycom  21900  htpyco1  21902  htpycc  21904  reparphti  21921  pcohtpylem  21943  pcorevlem  21950  cxpcn  23550  vmcn  26180  dipcn  26204  mndpluscn  28571  cvxscon  29754  cvmlift2lem6  29819
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