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Theorem tx1cn 20566
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 6773 . . 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 19886 . . . . . . . . . 10  |-  ( ( R  e.  (TopOn `  X )  /\  w  e.  R )  ->  w  C_  X )
43adantlr 719 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  w  C_  X )
5 xpss1 4905 . . . . . . . . 9  |-  ( w 
C_  X  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
64, 5syl 17 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
76sseld 3406 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
z  e.  ( w  X.  Y )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 639 . . . . . 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 5689 . . . . . . . 8  |-  ( ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> X  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5961 . . . . . . . 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 5839 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
1312eleq1d 2490 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 1st `  z
)  e.  w ) )
14 1st2nd2 6788 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp2nd 6782 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
16 elxp6 6783 . . . . . . . . . . . 12  |-  ( z  e.  ( w  X.  Y )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  w  /\  ( 2nd `  z )  e.  Y ) ) )
17 anass 653 . . . . . . . . . . . 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 805 . . . . . . . . . . . 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 276 . . . . . . . . . . 11  |-  ( z  e.  ( w  X.  Y )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  /\  ( 1st `  z )  e.  w
) )
2019baib 911 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  ->  ( z  e.  ( w  X.  Y
)  <->  ( 1st `  z
)  e.  w ) )
2114, 15, 20syl2anc 665 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( w  X.  Y )  <->  ( 1st `  z )  e.  w
) )
2213, 21bitr4d 259 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( w  X.  Y ) ) )
2322pm5.32i 641 . . . . . . 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 252 . . . . . 6  |-  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( w  X.  Y
) ) )
258, 24syl6rbbr 267 . . . . 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 2426 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  =  ( w  X.  Y
) )
27 toponmax 19885 . . . . . 6  |-  ( S  e.  (TopOn `  Y
)  ->  Y  e.  S )
2827ad2antlr 731 . . . . 5  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  Y  e.  S )
29 txopn 20559 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( w  e.  R  /\  Y  e.  S ) )  -> 
( w  X.  Y
)  e.  ( R 
tX  S ) )
3029anassrs 652 . . . . 5  |-  ( ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  /\  w  e.  R )  /\  Y  e.  S
)  ->  ( w  X.  Y )  e.  ( R  tX  S ) )
3128, 30mpdan 672 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y )  e.  ( R  tX  S ) )
3226, 31eqeltrd 2506 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2779 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 20548 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 simpl 458 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  R  e.  (TopOn `  X ) )
36 iscn 20193 . . 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 665 . 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 930 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2714    C_ wss 3379   <.cop 3947    X. cxp 4794   `'ccnv 4795    |` cres 4798   "cima 4799    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750  TopOnctopon 19860    Cn ccn 20182    tX ctx 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-map 7429  df-topgen 15285  df-top 19863  df-bases 19864  df-topon 19865  df-cn 20185  df-tx 20519
This theorem is referenced by:  txcn  20583  txcmpb  20601  cnmpt1st  20625  sxbrsiga  29064  txsconlem  29915  txscon  29916  hausgraph  36002
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