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Theorem ptpjpre2 20593
Description: The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptbas.1  |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  (
g `  y )  e.  ( F `  y
)  /\  E. z  e.  Fin  A. y  e.  ( A  \  z
) ( g `  y )  =  U. ( F `  y ) )  /\  x  = 
X_ y  e.  A  ( g `  y
) ) }
ptbasfi.2  |-  X  = 
X_ n  e.  A  U. ( F `  n
)
Assertion
Ref Expression
ptpjpre2  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  e.  B
)
Distinct variable groups:    B, n    w, g, x, y, n, I    z, g, A, n, w, x, y    U, g, n, w, x, y    g, F, n, w, x, y, z   
g, X, w, x, z    g, V, n, w, x, y, z
Allowed substitution hints:    B( x, y, z, w, g)    U( z)    I( z)    X( y, n)

Proof of Theorem ptpjpre2
StepHypRef Expression
1 ptbasfi.2 . . 3  |-  X  = 
X_ n  e.  A  U. ( F `  n
)
21ptpjpre1 20584 . 2  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  =  X_ n  e.  A  if ( n  =  I ,  U ,  U. ( F `  n )
) )
3 ptbas.1 . . 3  |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  (
g `  y )  e.  ( F `  y
)  /\  E. z  e.  Fin  A. y  e.  ( A  \  z
) ( g `  y )  =  U. ( F `  y ) )  /\  x  = 
X_ y  e.  A  ( g `  y
) ) }
4 simpll 758 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  A  e.  V )
5 snfi 7660 . . . 4  |-  { I }  e.  Fin
65a1i 11 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  { I }  e.  Fin )
7 simprr 764 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  U  e.  ( F `  I
) )
87ad2antrr 730 . . . . 5  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  U  e.  ( F `  I
) )
9 simpr 462 . . . . . 6  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  n  =  I )
109fveq2d 5885 . . . . 5  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  ( F `  n )  =  ( F `  I ) )
118, 10eleqtrrd 2510 . . . 4  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  U  e.  ( F `  n
) )
12 simplr 760 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  F : A --> Top )
1312ffvelrnda 6037 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  ( F `  n )  e.  Top )
14 eqid 2422 . . . . . . 7  |-  U. ( F `  n )  =  U. ( F `  n )
1514topopn 19934 . . . . . 6  |-  ( ( F `  n )  e.  Top  ->  U. ( F `  n )  e.  ( F `  n
) )
1613, 15syl 17 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  U. ( F `  n )  e.  ( F `  n
) )
1716adantr 466 . . . 4  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  -.  n  =  I )  ->  U. ( F `  n )  e.  ( F `  n
) )
1811, 17ifclda 3943 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  if (
n  =  I ,  U ,  U. ( F `  n )
)  e.  ( F `
 n ) )
19 eldifsni 4126 . . . . . 6  |-  ( n  e.  ( A  \  { I } )  ->  n  =/=  I
)
2019neneqd 2621 . . . . 5  |-  ( n  e.  ( A  \  { I } )  ->  -.  n  =  I )
2120adantl 467 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  ( A  \  { I } ) )  ->  -.  n  =  I
)
2221iffalsed 3922 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  ( A  \  { I } ) )  ->  if ( n  =  I ,  U ,  U. ( F `  n ) )  =  U. ( F `  n )
)
233, 4, 6, 18, 22elptr2 20587 . 2  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  X_ n  e.  A  if (
n  =  I ,  U ,  U. ( F `  n )
)  e.  B )
242, 23eqeltrd 2507 1  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   A.wral 2771   E.wrex 2772    \ cdif 3433   ifcif 3911   {csn 3998   U.cuni 4219    |-> cmpt 4482   `'ccnv 4852   "cima 4856    Fn wfn 5596   -->wf 5597   ` cfv 5601   X_cixp 7533   Fincfn 7580   Topctop 19915
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 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7193  df-ixp 7534  df-en 7581  df-fin 7584  df-top 19919
This theorem is referenced by:  ptbasfi  20594  ptpjcn  20624
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