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Theorem tx1cn 17319
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 6157 . . 3  |-  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X
21a1i 10 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y ) --> X )
3 toponss 16683 . . . . . . . . . 10  |-  ( ( R  e.  (TopOn `  X )  /\  w  e.  R )  ->  w  C_  X )
43adantlr 695 . . . . . . . . 9  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  w  C_  X )
5 xpss1 4811 . . . . . . . . 9  |-  ( w 
C_  X  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
64, 5syl 15 . . . . . . . 8  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y ) 
C_  ( X  X.  Y ) )
76sseld 3192 . . . . . . 7  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
z  e.  ( w  X.  Y )  -> 
z  e.  ( X  X.  Y ) ) )
87pm4.71rd 616 . . . . . 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 5405 . . . . . . . 8  |-  ( ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> X  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5661 . . . . . . . 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 9 . . . . . . 7  |-  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  ( ( 1st  |`  ( X  X.  Y ) ) `  z )  e.  w
) )
12 fvres 5558 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
1312eleq1d 2362 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 1st `  z
)  e.  w ) )
14 1st2nd2 6175 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp2nd 6166 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
16 elxp6 6167 . . . . . . . . . . . 12  |-  ( z  e.  ( w  X.  Y )  <->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  w  /\  ( 2nd `  z )  e.  Y ) ) )
17 anass 630 . . . . . . . . . . . 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 773 . . . . . . . . . . . 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 264 . . . . . . . . . . 11  |-  ( z  e.  ( w  X.  Y )  <->  ( (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  /\  ( 1st `  z )  e.  w
) )
2019baib 871 . . . . . . . . . 10  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( 2nd `  z )  e.  Y
)  ->  ( z  e.  ( w  X.  Y
)  <->  ( 1st `  z
)  e.  w ) )
2114, 15, 20syl2anc 642 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
z  e.  ( w  X.  Y )  <->  ( 1st `  z )  e.  w
) )
2213, 21bitr4d 247 . . . . . . . 8  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  z  e.  ( w  X.  Y ) ) )
2322pm5.32i 618 . . . . . . 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 240 . . . . . 6  |-  ( z  e.  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  <->  ( z  e.  ( X  X.  Y
)  /\  z  e.  ( w  X.  Y
) ) )
258, 24syl6rbbr 255 . . . . 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 2294 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  =  ( w  X.  Y
) )
27 toponmax 16682 . . . . . 6  |-  ( S  e.  (TopOn `  Y
)  ->  Y  e.  S )
2827ad2antlr 707 . . . . 5  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  Y  e.  S )
29 txopn 17313 . . . . . 6  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  ( w  e.  R  /\  Y  e.  S ) )  -> 
( w  X.  Y
)  e.  ( R 
tX  S ) )
3029anassrs 629 . . . . 5  |-  ( ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  /\  w  e.  R )  /\  Y  e.  S
)  ->  ( w  X.  Y )  e.  ( R  tX  S ) )
3128, 30mpdan 649 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  (
w  X.  Y )  e.  ( R  tX  S ) )
3226, 31eqeltrd 2370 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2639 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 17302 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
35 simpl 443 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  R  e.  (TopOn `  X ) )
36 iscn 16981 . . 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 642 . 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 888 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   <.cop 3656    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271
This theorem is referenced by:  txcn  17336  txcmpb  17354  cnmpt1st  17378  txsconlem  23786  txscon  23787  hausgraph  27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-tx 17273
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