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Theorem tx1cn 19313
Description: Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx1cn  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )

Proof of Theorem tx1cn
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 6707 . . 3  |-  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X
21a1i 11 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X )
3 toponss 18665 . . . . . . . . . 10  |-  ( ( R  e.  (TopOn `  X )  /\  w  e.  R )  ->  w  C_  X )
43adantlr 714 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  w  C_  X )
5 xpss1 5055 . . . . . . . . 9  |-  ( w 
C_  X  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
64, 5syl 16 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
76sseld 3462 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
z  e.  ( w  X.  Y )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 635 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
z  e.  ( w  X.  Y )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( w  X.  Y
) ) ) )
9 ffn 5666 . . . . . . . 8  |-  ( ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> X  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5931 . . . . . . . 8  |-  ( ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
)  ->  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y
) ) " w
)  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 1st  |`  ( X  X.  Y ) ) `  z )  e.  w
) ) )
111, 9, 10mp2b 10 . . . . . . 7  |-  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 1st  |`  ( X  X.  Y ) ) `  z )  e.  w
) )
12 fvres 5812 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
1312eleq1d 2523 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 1st `  z
)  e.  w ) )
14 1st2nd2 6722 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp2nd 6716 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
16 elxp6 6717 . . . . . . . . . . . 12  |-  ( z  e.  ( w  X.  Y )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  w  /\  ( 2nd `  z )  e.  Y ) ) )
17 anass 649 . . . . . . . . . . . 12  |-  ( ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  w
)  /\  ( 2nd `  z )  e.  Y
)  <->  ( z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  w  /\  ( 2nd `  z )  e.  Y ) ) )
18 an32 796 . . . . . . . . . . . 12  |-  ( ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 1st `  z )  e.  w
)  /\  ( 2nd `  z )  e.  Y
)  <->  ( ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  ( 2nd `  z )  e.  Y )  /\  ( 1st `  z )  e.  w ) )
1916, 17, 183bitr2i 273 . . . . . . . . . . 11  |-  ( z  e.  ( w  X.  Y )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  /\  ( 1st `  z )  e.  w
) )
2019baib 896 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  ->  ( z  e.  ( w  X.  Y
)  <->  ( 1st `  z
)  e.  w ) )
2114, 15, 20syl2anc 661 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( w  X.  Y )  <->  ( 1st `  z )  e.  w
) )
2213, 21bitr4d 256 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( w  X.  Y ) ) )
2322pm5.32i 637 . . . . . . 7  |-  ( ( z  e.  ( X  X.  Y )  /\  ( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( w  X.  Y
) ) )
2411, 23bitri 249 . . . . . 6  |-  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( w  X.  Y
) ) )
258, 24syl6rbbr 264 . . . . 5  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
z  e.  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  <->  z  e.  ( w  X.  Y
) ) )
2625eqrdv 2451 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  =  ( w  X.  Y
) )
27 toponmax 18664 . . . . . 6  |-  ( S  e.  (TopOn `  Y
)  ->  Y  e.  S )
2827ad2antlr 726 . . . . 5  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  Y  e.  S )
29 txopn 19306 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( w  e.  R  /\  Y  e.  S ) )  -> 
( w  X.  Y
)  e.  ( R 
tX  S ) )
3029anassrs 648 . . . . 5  |-  ( ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  /\  w  e.  R )  /\  Y  e.  S
)  ->  ( w  X.  Y )  e.  ( R  tX  S ) )
3128, 30mpdan 668 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y )  e.  ( R  tX  S ) )
3226, 31eqeltrd 2542 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2829 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 19295 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 simpl 457 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  R  e.  (TopOn `  X ) )
36 iscn 18970 . . 3  |-  ( ( ( R  tX  S
)  e.  (TopOn `  ( X  X.  Y
) )  /\  R  e.  (TopOn `  X )
)  ->  ( ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  R
)  <->  ( ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X  /\  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) ) ) )
3734, 35, 36syl2anc 661 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  R
)  <->  ( ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X  /\  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) ) ) )
382, 33, 37mpbir2and 913 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2798    C_ wss 3435   <.cop 3990    X. cxp 4945   `'ccnv 4946    |` cres 4949   "cima 4950    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685  TopOnctopon 18630    Cn ccn 18959    tX ctx 19264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-map 7325  df-topgen 14500  df-top 18634  df-bases 18636  df-topon 18637  df-cn 18962  df-tx 19266
This theorem is referenced by:  txcn  19330  txcmpb  19348  cnmpt1st  19372  sxbrsiga  26848  txsconlem  27272  txscon  27273  hausgraph  29727
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