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Theorem cnmpt2nd 19933
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 6805 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5797 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  2nd  Fn  _V
4 ssv 3524 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5694 . . . . 5  |-  ( ( 2nd  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 672 . . . 4  |-  ( 2nd  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5 5913 . . . 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 208 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5880 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
109mpteq2ia 4529 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 2nd `  z
) )
11 vex 3116 . . . . 5  |-  x  e. 
_V
12 vex 3116 . . . . 5  |-  y  e. 
_V
1311, 12op2ndd 6795 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
1413mpt2mpt 6378 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 2nd `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  y )
158, 10, 143eqtri 2500 . 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 19874 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  K ) )
1916, 17, 18syl2anc 661 . 2  |-  ( ph  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  K ) )
2015, 19syl5eqelr 2560 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 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476    |-> cmpt 4505    X. cxp 4997    |` cres 5001    Fn wfn 5583   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   2ndc2nd 6783  TopOnctopon 19190    Cn ccn 19519    tX ctx 19824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-topgen 14699  df-top 19194  df-bases 19196  df-topon 19197  df-cn 19522  df-tx 19826
This theorem is referenced by:  cnmptcom  19942  xkofvcn  19948  cnmptk2  19950  txhmeo  20067  txswaphmeo  20069  ptunhmeo  20072  xkohmeo  20079  tgpsubcn  20352  istgp2  20353  oppgtmd  20359  prdstmdd  20385  dvrcn  20449  divcn  21135  cnrehmeo  21216  htpycom  21239  htpyco1  21241  htpycc  21243  reparphti  21260  pcohtpylem  21282  pcorevlem  21289  cxpcn  22875  vmcn  25313  dipcn  25337  mndpluscn  27572  cvxscon  28356  cvmlift2lem6  28421
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