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Theorem ptcmpg 17767
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 17768). (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 2432 . . . 4  |-  F/_ k
( F `  a
)
3 nfcv 2432 . . . 4  |-  F/_ a
( F `  k
)
4 nfcv 2432 . . . 4  |-  F/_ k
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
5 nfcv 2432 . . . 4  |-  F/_ u
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
6 nfcv 2432 . . . 4  |-  F/_ a
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
7 nfcv 2432 . . . 4  |-  F/_ b
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
8 fveq2 5541 . . . 4  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
9 fveq2 5541 . . . . . . . 8  |-  ( a  =  k  ->  (
w `  a )  =  ( w `  k ) )
109mpteq2dv 4123 . . . . . . 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 4873 . . . . . 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 5027 . . . . 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 5024 . . . . 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 2348 . . . 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 5940 . . 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 5541 . . . . 5  |-  ( n  =  m  ->  ( F `  n )  =  ( F `  m ) )
1716unieqd 3854 . . . 4  |-  ( n  =  m  ->  U. ( F `  n )  =  U. ( F `  m ) )
1817cbvixpv 6850 . . 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 17138 . . . . . . . 8  |-  ( k  e.  Comp  ->  k  e. 
Top )
2221ssriv 3197 . . . . . . 7  |-  Comp  C_  Top
23 fss 5413 . . . . . . 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 17305 . . . . . 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 2346 . . . 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 2370 . . 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 17766 . 2  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  ( Xt_ `  F )  e.  Comp )
321, 31syl5eqel 2380 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271    e. cmpt2 5876   X_cixp 6833   cardccrd 7584   Xt_cpt 13359   Topctop 16647   Compccmp 17129  UFLcufl 17611
This theorem is referenced by:  ptcmp  17768  dfac21  27267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-wdom 7289  df-card 7588  df-acn 7591  df-topgen 13360  df-pt 13361  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cmp 17130  df-fbas 17536  df-fg 17537  df-fil 17557  df-ufil 17612  df-ufl 17613  df-flim 17650  df-fcls 17652
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