Metamath Proof Explorer

Theorem tx2cn

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)

Ref Expression
Assertion tx2cn 
|- ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )

Proof

Step Hyp Ref Expression
1 f2ndres 
 |-  ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y
2 1 a1i 
 |-  ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y )
3 ffn 
 |-  ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
4 elpreima 
 |-  ( ( 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 ) ) )
5 1 3 4 mp2b 
 |-  ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ ( ( 2nd |` ( X X. Y ) ) ` z ) e. w ) )
6 fvres 
 |-  ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
7 6 eleq1d 
 |-  ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> ( 2nd ` z ) e. w ) )
8 1st2nd2 
 |-  ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9 xp1st 
 |-  ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
10 elxp6 
 |-  ( z e. ( X X. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
11 anass 
 |-  ( ( ( 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 ) ) )
12 10 11 bitr4i 
 |-  ( z e. ( X X. w ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) /\ ( 2nd ` z ) e. w ) )
13 12 baib 
 |-  ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
14 8 9 13 syl2anc 
 |-  ( z e. ( X X. Y ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
15 7 14 bitr4d 
 |-  ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> z e. ( X X. w ) ) )
16 15 pm5.32i 
 |-  ( ( z e. ( X X. Y ) /\ ( ( 2nd |` ( X X. Y ) ) ` z ) e. w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
17 5 16 bitri 
 |-  ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
18 toponss 
 |-  ( ( S e. ( TopOn ` Y ) /\ w e. S ) -> w C_ Y )
19 18 adantll 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> w C_ Y )
20 xpss2 
 |-  ( w C_ Y -> ( X X. w ) C_ ( X X. Y ) )
21 19 20 syl 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) C_ ( X X. Y ) )
22 21 sseld 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( z e. ( X X. w ) -> z e. ( X X. Y ) ) )
23 22 pm4.71rd 
 |-  ( ( ( 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 ) ) ) )
24 17 23 bitr4id 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> z e. ( X X. w ) ) )
25 24 eqrdv 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) = ( X X. w ) )
26 toponmax 
 |-  ( R e. ( TopOn ` X ) -> X e. R )
27 txopn 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ ( X e. R /\ w e. S ) ) -> ( X X. w ) e. ( R tX S ) )
28 27 expr 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ X e. R ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
29 26 28 mpidan 
 |-  ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
30 29 imp 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) e. ( R tX S ) )
31 25 30 eqeltrd 
 |-  ( ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
32 31 ralrimiva 
 |-  ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33 txtopon 
 |-  ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( R tX S ) e. ( TopOn ` ( X X. Y ) ) )
34 iscn 
 |-  ( ( ( 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 ) ) ) )
35 33 34 sylancom 
 |-  ( ( 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 ) ) ) )
36 2 32 35 mpbir2and 
 |-  ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )