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Theorem ptcmpg 20741
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 20742). (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 2564 . . . 4  |-  F/_ k
( F `  a
)
3 nfcv 2564 . . . 4  |-  F/_ a
( F `  k
)
4 nfcv 2564 . . . 4  |-  F/_ k
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
5 nfcv 2564 . . . 4  |-  F/_ u
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  a ) ) "
b )
6 nfcv 2564 . . . 4  |-  F/_ a
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
7 nfcv 2564 . . . 4  |-  F/_ b
( `' ( w  e.  X_ n  e.  A  U. ( F `  n
)  |->  ( w `  k ) ) "
u )
8 fveq2 5805 . . . 4  |-  ( a  =  k  ->  ( F `  a )  =  ( F `  k ) )
9 fveq2 5805 . . . . . . . 8  |-  ( a  =  k  ->  (
w `  a )  =  ( w `  k ) )
109mpteq2dv 4481 . . . . . . 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 5120 . . . . . 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 5277 . . . . 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 5274 . . . . 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 2463 . . . 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 6312 . . 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 5805 . . . . 5  |-  ( n  =  m  ->  ( F `  n )  =  ( F `  m ) )
1716unieqd 4200 . . . 4  |-  ( n  =  m  ->  U. ( F `  n )  =  U. ( F `  m ) )
1817cbvixpv 7445 . . 3  |-  X_ n  e.  A  U. ( F `  n )  =  X_ m  e.  A  U. ( F `  m
)
19 simp1 997 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  A  e.  V )
20 simp2 998 . . 3  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Comp )
21 cmptop 20080 . . . . . . . 8  |-  ( k  e.  Comp  ->  k  e. 
Top )
2221ssriv 3445 . . . . . . 7  |-  Comp  C_  Top
23 fss 5678 . . . . . . 7  |-  ( ( F : A --> Comp  /\  Comp  C_ 
Top )  ->  F : A --> Top )
2420, 22, 23sylancl 660 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  F : A
--> Top )
251ptuni 20279 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> Top )  -> 
X_ n  e.  A  U. ( F `  n
)  =  U. J
)
2619, 24, 25syl2anc 659 . . . . 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 2461 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X_ n  e.  A  U. ( F `
 n )  =  X )
29 simp3 999 . . . 4  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  X  e.  (UFL  i^i  dom  card ) )
3028, 29eqeltrd 2490 . . 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 20740 . 2  |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card )
)  ->  ( Xt_ `  F )  e.  Comp )
321, 31syl5eqel 2494 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 974    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413   U.cuni 4190    |-> cmpt 4452   `'ccnv 4941   dom cdm 4942   "cima 4945   -->wf 5521   ` cfv 5525    |-> cmpt2 6236   X_cixp 7427   cardccrd 8268   Xt_cpt 14945   Topctop 19578   Compccmp 20071  UFLcufl 20585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-omul 7092  df-er 7268  df-map 7379  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fi 7825  df-wdom 7939  df-card 8272  df-acn 8275  df-topgen 14950  df-pt 14951  df-fbas 18628  df-fg 18629  df-top 19583  df-bases 19585  df-topon 19586  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-cmp 20072  df-fil 20531  df-ufil 20586  df-ufl 20587  df-flim 20624  df-fcls 20626
This theorem is referenced by:  ptcmp  20742  dfac21  35355
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