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Theorem tx2cn 17644
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 6371 . . 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 16996 . . . . . . . . . 10  |-  ( ( S  e.  (TopOn `  Y )  /\  w  e.  S )  ->  w  C_  Y )
43adantll 696 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  w  C_  Y )
5 xpss2 4987 . . . . . . . . 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 3349 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  (
z  e.  ( X  X.  w )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 618 . . . . . 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 5593 . . . . . . . 8  |-  ( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> Y  ->  ( 2nd  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5852 . . . . . . . 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 5747 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 2nd  |`  ( X  X.  Y ) ) `
 z )  =  ( 2nd `  z
) )
1312eleq1d 2504 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 2nd `  z
)  e.  w ) )
14 1st2nd2 6388 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp1st 6378 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
16 elxp6 6380 . . . . . . . . . . . 12  |-  ( z  e.  ( X  X.  w )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  X  /\  ( 2nd `  z )  e.  w ) ) )
17 anass 632 . . . . . . . . . . . 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 245 . . . . . . . . . . 11  |-  ( z  e.  ( X  X.  w )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  /\  ( 2nd `  z )  e.  w
) )
1918baib 873 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  X
)  ->  ( z  e.  ( X  X.  w
)  <->  ( 2nd `  z
)  e.  w ) )
2014, 15, 19syl2anc 644 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( X  X.  w )  <->  ( 2nd `  z )  e.  w
) )
2113, 20bitr4d 249 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 2nd  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( X  X.  w ) ) )
2221pm5.32i 620 . . . . . . 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 242 . . . . . 6  |-  ( z  e.  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( X  X.  w
) ) )
248, 23syl6rbbr 257 . . . . 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 2436 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  =  ( X  X.  w
) )
26 toponmax 16995 . . . . . . 7  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
2726adantr 453 . . . . . 6  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  X  e.  R )
28 txopn 17636 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( X  e.  R  /\  w  e.  S ) )  -> 
( X  X.  w
)  e.  ( R 
tX  S ) )
2928expr 600 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  X  e.  R )  ->  (
w  e.  S  -> 
( X  X.  w
)  e.  ( R 
tX  S ) ) )
3027, 29mpdan 651 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( w  e.  S  ->  ( X  X.  w )  e.  ( R  tX  S
) ) )
3130imp 420 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( X  X.  w )  e.  ( R  tX  S
) )
3225, 31eqeltrd 2512 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  S )  ->  ( `' ( 2nd  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2791 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  S  ( `' ( 2nd  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 17625 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 iscn 17301 . . 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 650 . 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 890 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   <.cop 3819    X. cxp 4878   `'ccnv 4879    |` cres 4882   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  TopOnctopon 16961    Cn ccn 17290    tX ctx 17594
This theorem is referenced by:  txcn  17660  txcmpb  17678  txkgen  17686  cnmpt2nd  17703  sxbrsiga  24642  txsconlem  24929  txscon  24930  hausgraph  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cn 17293  df-tx 17596
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