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Theorem ptcmpg 20285
Description: Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if  ~P ~P X is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 20286). (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
ptcmpg.1  |-  J  =  ( Xt_ `  F
)
ptcmpg.2  |-  X  = 
U. J
Assertion
Ref Expression
ptcmpg  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  J  e.  Comp )

Proof of Theorem ptcmpg
Dummy variables  a 
b  k  m  n  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmpg.1 . 2  |-  J  =  ( Xt_ `  F
)
2 nfcv 2622 . . . 4  |-  F/_ k
( F `  a
)
3 nfcv 2622 . . . 4  |-  F/_ a
( F `  k
)
4 nfcv 2622 . . . 4  |-  F/_ k
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
5 nfcv 2622 . . . 4  |-  F/_ u
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
6 nfcv 2622 . . . 4  |-  F/_ a
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
7 nfcv 2622 . . . 4  |-  F/_ b
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
8 fveq2 5857 . . . 4  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
9 fveq2 5857 . . . . . . . 8  |-  ( a  =  k  ->  (
w `  a )  =  ( w `  k ) )
109mpteq2dv 4527 . . . . . . 7  |-  ( a  =  k  ->  (
w  e.  X_ n  e.  A  U. ( F `  n )  |->  ( w `  a
) )  =  ( w  e.  X_ n  e.  A  U. ( F `  n )  |->  ( w `  k
) ) )
1110cnveqd 5169 . . . . . 6  |-  ( a  =  k  ->  `' ( w  e.  X_ n  e.  A  U. ( F `  n )  |->  ( w `  a
) )  =  `' ( w  e.  X_ n  e.  A  U. ( F `  n )  |->  ( w `  k
) ) )
1211imaeq1d 5327 . . . . 5  |-  ( a  =  k  ->  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )  =  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
b ) )
13 imaeq2 5324 . . . . 5  |-  ( b  =  u  ->  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
b )  =  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u ) )
1412, 13sylan9eq 2521 . . . 4  |-  ( ( a  =  k  /\  b  =  u )  ->  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )  =  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u ) )
152, 3, 4, 5, 6, 7, 8, 14cbvmpt2x 6350 . . 3  |-  ( a  e.  A ,  b  e.  ( F `  a )  |->  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n ) 
|->  ( w `  a
) ) " b
) )  =  ( k  e.  A ,  u  e.  ( F `  k )  |->  ( `' ( w  e.  X_ n  e.  A  U. ( F `  n ) 
|->  ( w `  k
) ) " u
) )
16 fveq2 5857 . . . . 5  |-  ( n  =  m  ->  ( F `  n )  =  ( F `  m ) )
1716unieqd 4248 . . . 4  |-  ( n  =  m  ->  U. ( F `  n )  =  U. ( F `  m ) )
1817cbvixpv 7477 . . 3  |-  X_ n  e.  A  U. ( F `  n )  =  X_ m  e.  A  U. ( F `  m
)
19 simp1 991 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  A  e.  V )
20 simp2 992 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Comp )
21 cmptop 19654 . . . . . . . 8  |-  ( k  e.  Comp  ->  k  e. 
Top )
2221ssriv 3501 . . . . . . 7  |-  Comp  C_  Top
23 fss 5730 . . . . . . 7  |-  ( ( F : A --> Comp  /\  Comp  C_ 
Top )  ->  F : A --> Top )
2420, 22, 23sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Top )
251ptuni 19823 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Top )  -> 
X_ n  e.  A  U. ( F `  n
)  =  U. J
)
2619, 24, 25syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  = 
U. J )
27 ptcmpg.2 . . . . 5  |-  X  = 
U. J
2826, 27syl6eqr 2519 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  =  X )
29 simp3 993 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X  e.  (UFL  i^i  dom  card ) )
3028, 29eqeltrd 2548 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  e.  (UFL  i^i  dom  card )
)
3115, 18, 19, 20, 30ptcmplem5 20284 . 2  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  ( Xt_ `  F )  e.  Comp )
321, 31syl5eqel 2552 1  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  J  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   U.cuni 4238    |-> cmpt 4498   `'ccnv 4991   dom cdm 4992   "cima 4995   -->wf 5575   ` cfv 5579    |-> cmpt2 6277   X_cixp 7459   cardccrd 8305   Xt_cpt 14683   Topctop 19154   Compccmp 19645  UFLcufl 20129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  df-wdom 7974  df-card 8309  df-acn 8312  df-topgen 14688  df-pt 14689  df-fbas 18180  df-fg 18181  df-top 19159  df-bases 19161  df-topon 19162  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-cmp 19646  df-fil 20075  df-ufil 20130  df-ufl 20131  df-flim 20168  df-fcls 20170
This theorem is referenced by:  ptcmp  20286  dfac21  30605
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