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Theorem cnmpt2nd 12300
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 6010 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5305 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 7 . . . . 5  |-  2nd  Fn  _V
4 ssv 3085 . . . . 5  |-  ( X  X.  Y )  C_  _V
5 fnssres 5194 . . . . 5  |-  ( ( 2nd  Fn  _V  /\  ( X  X.  Y
)  C_  _V )  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y ) )
63, 4, 5mp2an 420 . . . 4  |-  ( 2nd  |`  ( X  X.  Y
) )  Fn  ( X  X.  Y )
7 dffn5im 5421 . . . 4  |-  ( ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
)  ->  ( 2nd  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `
 z ) ) )
86, 7ax-mp 7 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) )  =  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `
 z ) )
9 fvres 5399 . . . 4  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
109mpteq2ia 3974 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( ( 2nd  |`  ( X  X.  Y ) ) `  z ) )  =  ( z  e.  ( X  X.  Y ) 
|->  ( 2nd `  z
) )
11 vex 2660 . . . . 5  |-  x  e. 
_V
12 vex 2660 . . . . 5  |-  y  e. 
_V
1311, 12op2ndd 6001 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
1413mpompt 5817 . . 3  |-  ( z  e.  ( X  X.  Y )  |->  ( 2nd `  z ) )  =  ( x  e.  X ,  y  e.  Y  |->  y )
158, 10, 143eqtri 2139 . 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 12281 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( J  tX  K
)  Cn  K ) )
1916, 17, 18syl2anc 406 . 2  |-  ( ph  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( J 
tX  K )  Cn  K ) )
2015, 19syl5eqelr 2202 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J  tX  K
)  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2657    C_ wss 3037    |-> cmpt 3949    X. cxp 4497    |` cres 4501    Fn wfn 5076   -onto->wfo 5079   ` cfv 5081  (class class class)co 5728    e. cmpo 5730   2ndc2nd 5991  TopOnctopon 12020    Cn ccn 12197    tX ctx 12263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-map 6498  df-topgen 11984  df-top 12008  df-topon 12021  df-bases 12053  df-cn 12200  df-tx 12264
This theorem is referenced by:  cnmptcom  12309  divcnap  12541
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