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Theorem tx1cn 19842
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 6803 . . 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 19194 . . . . . . . . . 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 5109 . . . . . . . . 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 3503 . . . . . . 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 5729 . . . . . . . 8  |-  ( ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y
) --> X  ->  ( 1st  |`  ( X  X.  Y ) )  Fn  ( X  X.  Y
) )
10 elpreima 5999 . . . . . . . 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 5878 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  (
( 1st  |`  ( X  X.  Y ) ) `
 z )  =  ( 1st `  z
) )
1312eleq1d 2536 . . . . . . . . 9  |-  ( z  e.  ( X  X.  Y )  ->  (
( ( 1st  |`  ( X  X.  Y ) ) `
 z )  e.  w  <->  ( 1st `  z
)  e.  w ) )
14 1st2nd2 6818 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
15 xp2nd 6812 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
16 elxp6 6813 . . . . . . . . . . . 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 901 . . . . . . . . . 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 2464 . . . 4  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  =  ( w  X.  Y
) )
27 toponmax 19193 . . . . . 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 19835 . . . . . 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 2555 . . 3  |-  ( ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  /\  w  e.  R )  ->  ( `' ( 1st  |`  ( X  X.  Y ) )
" w )  e.  ( R  tX  S
) )
3332ralrimiva 2878 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  A. w  e.  R  ( `' ( 1st  |`  ( X  X.  Y ) ) "
w )  e.  ( R  tX  S ) )
34 txtopon 19824 . . 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 19499 . . 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 920 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 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   <.cop 4033    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780  TopOnctopon 19159    Cn ccn 19488    tX ctx 19793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-topgen 14692  df-top 19163  df-bases 19165  df-topon 19166  df-cn 19491  df-tx 19795
This theorem is referenced by:  txcn  19859  txcmpb  19877  cnmpt1st  19901  sxbrsiga  27898  txsconlem  28322  txscon  28323  hausgraph  30777
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