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Theorem ptpjpre2 17614
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 17605 . 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 732 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  A  e.  V )
5 snfi 7189 . . . 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 735 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  U  e.  ( F `  I
) )
87ad2antrr 708 . . . . 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 449 . . . . . 6  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  n  =  I )
109fveq2d 5734 . . . . 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 2515 . . . 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 733 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  F : A --> Top )
1312ffvelrnda 5872 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  ( F `  n )  e.  Top )
14 eqid 2438 . . . . . . 7  |-  U. ( F `  n )  =  U. ( F `  n )
1514topopn 16981 . . . . . 6  |-  ( ( F `  n )  e.  Top  ->  U. ( F `  n )  e.  ( F `  n
) )
1613, 15syl 16 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  U. ( F `  n )  e.  ( F `  n
) )
1716adantr 453 . . . 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 3768 . . 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 3930 . . . . . 6  |-  ( n  e.  ( A  \  { I } )  ->  n  =/=  I
)
2019neneqd 2619 . . . . 5  |-  ( n  e.  ( A  \  { I } )  ->  -.  n  =  I )
2120adantl 454 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  ( A  \  { I } ) )  ->  -.  n  =  I
)
22 iffalse 3748 . . . 4  |-  ( -.  n  =  I  ->  if ( n  =  I ,  U ,  U. ( F `  n ) )  =  U. ( F `  n )
)
2321, 22syl 16 . . 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 )
)
243, 4, 6, 18, 23elptr2 17608 . 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 )
252, 24eqeltrd 2512 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    \ cdif 3319   ifcif 3741   {csn 3816   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456   X_cixp 7065   Fincfn 7111   Topctop 16960
This theorem is referenced by:  ptbasfi  17615  ptpjcn  17645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-ixp 7066  df-en 7112  df-fin 7115  df-top 16965
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