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Theorem tx2cn 20196
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 6722 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y
21a1i 11 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y )
3 toponss 19515 . . . . . . . . . 10  |-  ( ( S  e.  (TopOn `  Y )  /\  w  e.  S )  ->  w  C_  Y )
43adantll 711 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  w  C_  Y )
5 xpss2 5025 . . . . . . . . 9  |-  ( w 
C_  Y  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
64, 5syl 16 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
76sseld 3416 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 633 . . . . . 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 5639 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> Y  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5909 . . . . . . . 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 10 . . . . . . 7  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 2nd  |`  ( X  X.  Y ) ) `  z )  e.  w
) )
12 fvres 5788 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
1312eleq1d 2451 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 2nd `  z
)  e.  w ) )
14 1st2nd2 6736 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp1st 6729 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
16 elxp6 6731 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  w )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
17 anass 647 . . . . . . . . . . . 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 252 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  w )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  /\  ( 2nd `  z )  e.  w
) )
1918baib 901 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  ->  ( z  e.  ( X  X.  w
)  <->  ( 2nd `  z
)  e.  w ) )
2014, 15, 19syl2anc 659 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( X  X.  w )  <->  ( 2nd `  z )  e.  w
) )
2113, 20bitr4d 256 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( X  X.  w ) ) )
2221pm5.32i 635 . . . . . . 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 249 . . . . . 6  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) )
248, 23syl6rbbr 264 . . . . 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 2379 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  =  ( X  X.  w
) )
26 toponmax 19514 . . . . . . 7  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
2726adantr 463 . . . . . 6  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  X  e.  R )
28 txopn 20188 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( X  e.  R  /\  w  e.  S ) )  -> 
( X  X.  w
)  e.  ( R 
tX  S ) )
2928expr 613 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  X  e.  R )  ->  (
w  e.  S  -> 
( X  X.  w
)  e.  ( R 
tX  S ) ) )
3027, 29mpdan 666 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( w  e.  S  ->  ( X  X.  w )  e.  ( R  tX  S
) ) )
3130imp 427 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  e.  ( R  tX  S
) )
3225, 31eqeltrd 2470 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2796 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 20177 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 iscn 19822 . . 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 ) ) ) )
3634, 35sylancom 665 . 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 ) ) ) )
372, 33, 36mpbir2and 920 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389   <.cop 3950    X. cxp 4911   `'ccnv 4912    |` cres 4915   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698  TopOnctopon 19480    Cn ccn 19811    tX ctx 20146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-topgen 14851  df-top 19484  df-bases 19486  df-topon 19487  df-cn 19814  df-tx 20148
This theorem is referenced by:  txcn  20212  txcmpb  20230  txkgen  20238  cnmpt2nd  20255  sxbrsiga  28417  txsconlem  28874  txscon  28875  hausgraph  31340
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