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Theorem ptcmpg 19648
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 19649). (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 2589 . . . 4  |-  F/_ k
( F `  a
)
3 nfcv 2589 . . . 4  |-  F/_ a
( F `  k
)
4 nfcv 2589 . . . 4  |-  F/_ k
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
5 nfcv 2589 . . . 4  |-  F/_ u
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
6 nfcv 2589 . . . 4  |-  F/_ a
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
7 nfcv 2589 . . . 4  |-  F/_ b
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
8 fveq2 5710 . . . 4  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
9 fveq2 5710 . . . . . . . 8  |-  ( a  =  k  ->  (
w `  a )  =  ( w `  k ) )
109mpteq2dv 4398 . . . . . . 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 5034 . . . . . 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 5187 . . . . 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 5184 . . . . 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 2495 . . . 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 6183 . . 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 5710 . . . . 5  |-  ( n  =  m  ->  ( F `  n )  =  ( F `  m ) )
1716unieqd 4120 . . . 4  |-  ( n  =  m  ->  U. ( F `  n )  =  U. ( F `  m ) )
1817cbvixpv 7300 . . 3  |-  X_ n  e.  A  U. ( F `  n )  =  X_ m  e.  A  U. ( F `  m
)
19 simp1 988 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  A  e.  V )
20 simp2 989 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Comp )
21 cmptop 19017 . . . . . . . 8  |-  ( k  e.  Comp  ->  k  e. 
Top )
2221ssriv 3379 . . . . . . 7  |-  Comp  C_  Top
23 fss 5586 . . . . . . 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 19186 . . . . . 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 2493 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  =  X )
29 simp3 990 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X  e.  (UFL  i^i  dom  card ) )
3028, 29eqeltrd 2517 . . 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 19647 . 2  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  ( Xt_ `  F )  e.  Comp )
321, 31syl5eqel 2527 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 965    = wceq 1369    e. wcel 1756    i^i cin 3346    C_ wss 3347   U.cuni 4110    e. cmpt 4369   `'ccnv 4858   dom cdm 4859   "cima 4862   -->wf 5433   ` cfv 5437    e. cmpt2 6112   X_cixp 7282   cardccrd 8124   Xt_cpt 14396   Topctop 18517   Compccmp 19008  UFLcufl 19492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-omul 6944  df-er 7120  df-map 7235  df-ixp 7283  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fi 7680  df-wdom 7793  df-card 8128  df-acn 8131  df-topgen 14401  df-pt 14402  df-fbas 17833  df-fg 17834  df-top 18522  df-bases 18524  df-topon 18525  df-cld 18642  df-ntr 18643  df-cls 18644  df-nei 18721  df-cmp 19009  df-fil 19438  df-ufil 19493  df-ufl 19494  df-flim 19531  df-fcls 19533
This theorem is referenced by:  ptcmp  19649  dfac21  29442
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