MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tx2cn Unicode version

Theorem tx2cn 17304
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 6142 . . 3  |-  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y
21a1i 10 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> Y )
3 toponss 16667 . . . . . . . . . 10  |-  ( ( S  e.  (TopOn `  Y )  /\  w  e.  S )  ->  w  C_  Y )
43adantll 694 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  w  C_  Y )
5 xpss2 4796 . . . . . . . . 9  |-  ( w 
C_  Y  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
64, 5syl 15 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  C_  ( X  X.  Y
) )
76sseld 3179 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 616 . . . . . 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 5389 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> Y  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5645 . . . . . . . 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 9 . . . . . . 7  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 2nd  |`  ( X  X.  Y ) ) `  z )  e.  w
) )
12 fvres 5542 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
1312eleq1d 2349 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 2nd `  z
)  e.  w ) )
14 1st2nd2 6159 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp1st 6149 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
16 elxp6 6151 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  w )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
17 anass 630 . . . . . . . . . . . 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 243 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  w )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  /\  ( 2nd `  z )  e.  w
) )
1918baib 871 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  ->  ( z  e.  ( X  X.  w
)  <->  ( 2nd `  z
)  e.  w ) )
2014, 15, 19syl2anc 642 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( X  X.  w )  <->  ( 2nd `  z )  e.  w
) )
2113, 20bitr4d 247 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( X  X.  w ) ) )
2221pm5.32i 618 . . . . . . 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 240 . . . . . 6  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) )
248, 23syl6rbbr 255 . . . . 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 2281 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  =  ( X  X.  w
) )
26 toponmax 16666 . . . . . . 7  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
2726adantr 451 . . . . . 6  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  X  e.  R )
28 txopn 17297 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( X  e.  R  /\  w  e.  S ) )  -> 
( X  X.  w
)  e.  ( R 
tX  S ) )
2928expr 598 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  X  e.  R )  ->  (
w  e.  S  -> 
( X  X.  w
)  e.  ( R 
tX  S ) ) )
3027, 29mpdan 649 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( w  e.  S  ->  ( X  X.  w )  e.  ( R  tX  S
) ) )
3130imp 418 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  e.  ( R  tX  S
) )
3225, 31eqeltrd 2357 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2626 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 17286 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 iscn 16965 . . 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 648 . 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 888 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   <.cop 3643    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255
This theorem is referenced by:  txcn  17320  txcmpb  17338  txkgen  17346  cnmpt2nd  17363  txsconlem  23771  txscon  23772  hausgraph  27531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257
  Copyright terms: Public domain W3C validator