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Theorem tx2cn 12345
Description: Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx2cn  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )

Proof of Theorem tx2cn
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f2ndres 6024 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y
21a1i 9 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y )
3 toponss 12099 . . . . . . . . . 10  |-  ( ( S  e.  (TopOn `  Y )  /\  w  e.  S )  ->  w  C_  Y )
43adantll 465 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  w  C_  Y )
5 xpss2 4618 . . . . . . . . 9  |-  ( w 
C_  Y  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
64, 5syl 14 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
76sseld 3064 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 389 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) ) )
9 ffn 5240 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> Y  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5505 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
)  ->  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y
) ) " w
)  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 2nd  |`  ( X  X.  Y ) ) `  z )  e.  w
) ) )
111, 9, 10mp2b 8 . . . . . . 7  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 2nd  |`  ( X  X.  Y ) ) `  z )  e.  w
) )
12 fvres 5411 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
1312eleq1d 2184 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 2nd `  z
)  e.  w ) )
14 1st2nd2 6039 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp1st 6029 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
16 elxp6 6033 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  w )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
17 anass 396 . . . . . . . . . . . 12  |-  ( ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  /\  ( 2nd `  z )  e.  w
)  <->  ( z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
1816, 17bitr4i 186 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  w )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  /\  ( 2nd `  z )  e.  w
) )
1918baib 887 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  ->  ( z  e.  ( X  X.  w
)  <->  ( 2nd `  z
)  e.  w ) )
2014, 15, 19syl2anc 406 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( X  X.  w )  <->  ( 2nd `  z )  e.  w
) )
2113, 20bitr4d 190 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( X  X.  w ) ) )
2221pm5.32i 447 . . . . . . 7  |-  ( ( z  e.  ( X  X.  Y )  /\  ( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) )
2311, 22bitri 183 . . . . . 6  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) )
248, 23syl6rbbr 198 . . . . 5  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  <->  z  e.  ( X  X.  w
) ) )
2524eqrdv 2113 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  =  ( X  X.  w
) )
26 toponmax 12098 . . . . . 6  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
27 txopn 12340 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( X  e.  R  /\  w  e.  S ) )  -> 
( X  X.  w
)  e.  ( R 
tX  S ) )
2827expr 370 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  X  e.  R )  ->  (
w  e.  S  -> 
( X  X.  w
)  e.  ( R 
tX  S ) ) )
2926, 28mpidan 417 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( w  e.  S  ->  ( X  X.  w )  e.  ( R  tX  S
) ) )
3029imp 123 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  e.  ( R  tX  S
) )
3125, 30eqeltrd 2192 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3231ralrimiva 2480 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
33 txtopon 12337 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
34 iscn 12272 . . 3  |-  ( ( ( R  tX  S
)  e.  (TopOn `  ( X  X.  Y
) )  /\  S  e.  (TopOn `  Y )
)  ->  ( ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  S
)  <->  ( ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y  /\  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) ) ) )
3533, 34sylancom 414 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  S
)  <->  ( ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y  /\  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) ) ) )
362, 32, 35mpbir2and 911 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391    C_ wss 3039   <.cop 3498    X. cxp 4505   `'ccnv 4506    |` cres 4509   "cima 4510    Fn wfn 5086   -->wf 5087   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003  TopOnctopon 12083    Cn ccn 12260    tX ctx 12327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 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 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
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 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-map 6510  df-topgen 12047  df-top 12071  df-topon 12084  df-bases 12116  df-cn 12263  df-tx 12328
This theorem is referenced by:  txcn  12350  cnmpt2nd  12364
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