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Theorem ptpjpre2 17331
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 17322 . 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 730 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  A  e.  V )
5 snfi 6984 . . . 4  |-  { I }  e.  Fin
65a1i 10 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  { I }  e.  Fin )
7 simprr 733 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  U  e.  ( F `  I
) )
87ad2antrr 706 . . . . 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 447 . . . . . 6  |-  ( ( ( ( ( A  e.  V  /\  F : A --> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  /\  n  =  I )  ->  n  =  I )
109fveq2d 5567 . . . . 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 2393 . . . 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 731 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `  I )
) )  ->  F : A --> Top )
13 ffvelrn 5701 . . . . . . 7  |-  ( ( F : A --> Top  /\  n  e.  A )  ->  ( F `  n
)  e.  Top )
1412, 13sylan 457 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  ( F `  n )  e.  Top )
15 eqid 2316 . . . . . . 7  |-  U. ( F `  n )  =  U. ( F `  n )
1615topopn 16708 . . . . . 6  |-  ( ( F `  n )  e.  Top  ->  U. ( F `  n )  e.  ( F `  n
) )
1714, 16syl 15 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  A
)  ->  U. ( F `  n )  e.  ( F `  n
) )
1817adantr 451 . . . 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
) )
1911, 18ifclda 3626 . . 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 ) )
20 eldifsni 3784 . . . . . 6  |-  ( n  e.  ( A  \  { I } )  ->  n  =/=  I
)
2120neneqd 2495 . . . . 5  |-  ( n  e.  ( A  \  { I } )  ->  -.  n  =  I )
2221adantl 452 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> Top )  /\  (
I  e.  A  /\  U  e.  ( F `  I ) ) )  /\  n  e.  ( A  \  { I } ) )  ->  -.  n  =  I
)
23 iffalse 3606 . . . 4  |-  ( -.  n  =  I  ->  if ( n  =  I ,  U ,  U. ( F `  n ) )  =  U. ( F `  n )
)
2422, 23syl 15 . . 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 )
)
253, 4, 6, 19, 24elptr2 17325 . 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 )
262, 25eqeltrd 2390 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 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578    \ cdif 3183   ifcif 3599   {csn 3674   U.cuni 3864    e. cmpt 4114   `'ccnv 4725   "cima 4729    Fn wfn 5287   -->wf 5288   ` cfv 5292   X_cixp 6860   Fincfn 6906   Topctop 16687
This theorem is referenced by:  ptbasfi  17332  ptpjcn  17361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1o 6521  df-ixp 6861  df-en 6907  df-fin 6910  df-top 16692
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