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Theorem ptcmpg 17751
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 17752). (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 2419 . . . 4  |-  F/_ k
( F `  a
)
3 nfcv 2419 . . . 4  |-  F/_ a
( F `  k
)
4 nfcv 2419 . . . 4  |-  F/_ k
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
5 nfcv 2419 . . . 4  |-  F/_ u
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
6 nfcv 2419 . . . 4  |-  F/_ a
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
7 nfcv 2419 . . . 4  |-  F/_ b
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
8 fveq2 5525 . . . 4  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
9 fveq2 5525 . . . . . . . 8  |-  ( a  =  k  ->  (
w `  a )  =  ( w `  k ) )
109mpteq2dv 4107 . . . . . . 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 4857 . . . . . 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 5011 . . . . 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 5008 . . . . 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 2335 . . . 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 5924 . . 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 5525 . . . . 5  |-  ( n  =  m  ->  ( F `  n )  =  ( F `  m ) )
1716unieqd 3838 . . . 4  |-  ( n  =  m  ->  U. ( F `  n )  =  U. ( F `  m ) )
1817cbvixpv 6834 . . 3  |-  X_ n  e.  A  U. ( F `  n )  =  X_ m  e.  A  U. ( F `  m
)
19 simp1 955 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  A  e.  V )
20 simp2 956 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Comp )
21 cmptop 17122 . . . . . . . 8  |-  ( k  e.  Comp  ->  k  e. 
Top )
2221ssriv 3184 . . . . . . 7  |-  Comp  C_  Top
23 fss 5397 . . . . . . 7  |-  ( ( F : A --> Comp  /\  Comp  C_ 
Top )  ->  F : A --> Top )
2420, 22, 23sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Top )
251ptuni 17289 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Top )  -> 
X_ n  e.  A  U. ( F `  n
)  =  U. J
)
2619, 24, 25syl2anc 642 . . . . 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 2333 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  =  X )
29 simp3 957 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X  e.  (UFL  i^i  dom  card ) )
3028, 29eqeltrd 2357 . . 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 17750 . 2  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  ( Xt_ `  F )  e.  Comp )
321, 31syl5eqel 2367 1  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  J  e.  Comp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255    e. cmpt2 5860   X_cixp 6817   cardccrd 7568   Xt_cpt 13343   Topctop 16631   Compccmp 17113  UFLcufl 17595
This theorem is referenced by:  ptcmp  17752  dfac21  27164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-wdom 7273  df-card 7572  df-acn 7575  df-topgen 13344  df-pt 13345  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cmp 17114  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596  df-ufl 17597  df-flim 17634  df-fcls 17636
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