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Theorem tx2cn 19841
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 6799 . . 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 19192 . . . . . . . . . 10  |-  ( ( S  e.  (TopOn `  Y )  /\  w  e.  S )  ->  w  C_  Y )
43adantll 713 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  w  C_  Y )
5 xpss2 5105 . . . . . . . . 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 3498 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 635 . . . . . 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 5724 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> Y  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5994 . . . . . . . 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 5873 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
1312eleq1d 2531 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 2nd `  z
)  e.  w ) )
14 1st2nd2 6813 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp1st 6806 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
16 elxp6 6808 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  w )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
17 anass 649 . . . . . . . . . . . 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 898 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  ->  ( z  e.  ( X  X.  w
)  <->  ( 2nd `  z
)  e.  w ) )
2014, 15, 19syl2anc 661 . . . . . . . . 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 637 . . . . . . 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 2459 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  =  ( X  X.  w
) )
26 toponmax 19191 . . . . . . 7  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
2726adantr 465 . . . . . 6  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  X  e.  R )
28 txopn 19833 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( X  e.  R  /\  w  e.  S ) )  -> 
( X  X.  w
)  e.  ( R 
tX  S ) )
2928expr 615 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  X  e.  R )  ->  (
w  e.  S  -> 
( X  X.  w
)  e.  ( R 
tX  S ) ) )
3027, 29mpdan 668 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( w  e.  S  ->  ( X  X.  w )  e.  ( R  tX  S
) ) )
3130imp 429 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  e.  ( R  tX  S
) )
3225, 31eqeltrd 2550 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2873 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 19822 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 iscn 19497 . . 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 667 . 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 915 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 369    = wceq 1374    e. wcel 1762   A.wral 2809    C_ wss 3471   <.cop 4028    X. cxp 4992   `'ccnv 4993    |` cres 4996   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775  TopOnctopon 19157    Cn ccn 19486    tX ctx 19791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-topgen 14690  df-top 19161  df-bases 19163  df-topon 19164  df-cn 19489  df-tx 19793
This theorem is referenced by:  txcn  19857  txcmpb  19875  txkgen  19883  cnmpt2nd  19900  sxbrsiga  27889  txsconlem  28313  txscon  28314  hausgraph  30768
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