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Theorem ptpjpre2 20371
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 20362 . 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 752 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  A  e.  V )
5 snfi 7633 . . . 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 758 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  U  e.  ( F `  I
) )
87ad2antrr 724 . . . . 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 459 . . . . . 6  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  n  =  I )
109fveq2d 5852 . . . . 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 2493 . . . 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 754 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  F : A --> Top )
1312ffvelrnda 6008 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  ( F `  n )  e.  Top )
14 eqid 2402 . . . . . . 7  |-  U. ( F `  n )  =  U. ( F `  n )
1514topopn 19705 . . . . . 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 463 . . . 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 3916 . . 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 4097 . . . . . 6  |-  ( n  e.  ( A  \  { I } )  ->  n  =/=  I
)
2019neneqd 2605 . . . . 5  |-  ( n  e.  ( A  \  { I } )  ->  -.  n  =  I )
2120adantl 464 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  ( A  \  { I } ) )  ->  -.  n  =  I
)
2221iffalsed 3895 . . 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 20365 . 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 2490 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 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754    \ cdif 3410   ifcif 3884   {csn 3971   U.cuni 4190    |-> cmpt 4452   `'ccnv 4821   "cima 4825    Fn wfn 5563   -->wf 5564   ` cfv 5568   X_cixp 7506   Fincfn 7553   Topctop 19684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-1o 7166  df-ixp 7507  df-en 7554  df-fin 7557  df-top 19689
This theorem is referenced by:  ptbasfi  20372  ptpjcn  20402
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