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Theorem cnmpt2nd 19242
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 6597 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5622 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  2nd  Fn  _V
4 ssv 3376 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5524 . . . . 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 5737 . . . 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 5704 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
109mpteq2ia 4374 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 2nd `  z
) )
11 vex 2975 . . . . 5  |-  x  e. 
_V
12 vex 2975 . . . . 5  |-  y  e. 
_V
1311, 12op2ndd 6588 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
1413mpt2mpt 6182 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 2nd `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  y )
158, 10, 143eqtri 2467 . 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 19183 . . 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 2528 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 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328    e. cmpt 4350    X. cxp 4838    |` cres 4842    Fn wfn 5413   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   2ndc2nd 6576  TopOnctopon 18499    Cn ccn 18828    tX ctx 19133
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216  df-topgen 14382  df-top 18503  df-bases 18505  df-topon 18506  df-cn 18831  df-tx 19135
This theorem is referenced by:  cnmptcom  19251  xkofvcn  19257  cnmptk2  19259  txhmeo  19376  txswaphmeo  19378  ptunhmeo  19381  xkohmeo  19388  tgpsubcn  19661  istgp2  19662  oppgtmd  19668  prdstmdd  19694  dvrcn  19758  divcn  20444  cnrehmeo  20525  htpycom  20548  htpyco1  20550  htpycc  20552  reparphti  20569  pcohtpylem  20591  pcorevlem  20598  cxpcn  22183  vmcn  24094  dipcn  24118  mndpluscn  26356  cvxscon  27132  cvmlift2lem6  27197
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