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Theorem hausmapdom 17226
Description: If  X is a first-countable Hausdorff space, then the cardinality of the closure of a set  A is bounded by  NN to the power  A. In particular, a first-countable Hausdorff space with a dense subset  A has cardinality at most  A ^ NN, and a separable first-countable Hausdorff space has cardinality at most  ~P NN. (Compare hauspwpwdom 17683 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
hauspwdom.1  |-  X  = 
U. J
Assertion
Ref Expression
hausmapdom  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )

Proof of Theorem hausmapdom
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hauspwdom.1 . . . . . . . 8  |-  X  = 
U. J
211stcelcls 17187 . . . . . . 7  |-  ( ( J  e.  1stc  /\  A  C_  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  E. f
( f : NN --> A  /\  f ( ~~> t `  J ) x ) ) )
323adant1 973 . . . . . 6  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  E. f ( f : NN --> A  /\  f
( ~~> t `  J
) x ) ) )
4 simp3 957 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  C_  X
)
5 uniexg 4517 . . . . . . . . . . . 12  |-  ( J  e.  Haus  ->  U. J  e.  _V )
653ad2ant1 976 . . . . . . . . . . 11  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  U. J  e. 
_V )
71, 6syl5eqel 2367 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  X  e.  _V )
8 ssexg 4160 . . . . . . . . . 10  |-  ( ( A  C_  X  /\  X  e.  _V )  ->  A  e.  _V )
94, 7, 8syl2anc 642 . . . . . . . . 9  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  e.  _V )
10 nnex 9752 . . . . . . . . 9  |-  NN  e.  _V
11 elmapg 6785 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  NN  e.  _V )  -> 
( f  e.  ( A  ^m  NN )  <-> 
f : NN --> A ) )
129, 10, 11sylancl 643 . . . . . . . 8  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( f  e.  ( A  ^m  NN ) 
<->  f : NN --> A ) )
1312anbi1d 685 . . . . . . 7  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( (
f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x )  <-> 
( f : NN --> A  /\  f ( ~~> t `  J ) x ) ) )
1413exbidv 1612 . . . . . 6  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( E. f ( f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x )  <->  E. f ( f : NN --> A  /\  f
( ~~> t `  J
) x ) ) )
153, 14bitr4d 247 . . . . 5  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  E. f ( f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x ) ) )
16 df-rex 2549 . . . . 5  |-  ( E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x  <->  E. f
( f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x ) )
1715, 16syl6bbr 254 . . . 4  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x ) )
18 vex 2791 . . . . 5  |-  x  e. 
_V
1918elima 5017 . . . 4  |-  ( x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  <->  E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x )
2017, 19syl6bbr 254 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) ) ) )
2120eqrdv 2281 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  =  ( ( ~~> t `  J
) " ( A  ^m  NN ) ) )
22 ovex 5883 . . 3  |-  ( A  ^m  NN )  e. 
_V
23 lmfun 17109 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
24233ad2ant1 976 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  Fun  ( ~~> t `  J ) )
25 imadomg 8159 . . 3  |-  ( ( A  ^m  NN )  e.  _V  ->  ( Fun  ( ~~> t `  J
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) ) )
2622, 24, 25mpsyl 59 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) )
2721, 26eqbrtrd 4043 1  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ~<_ cdom 6861   NNcn 9746   clsccl 16755   ~~> tclm 16956   Hauscha 17036   1stcc1stc 17163
This theorem is referenced by:  hauspwdom  17227
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  ax-inf2 7342  ax-cc 8061  ax-ac2 8089  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-acn 7575  df-ac 7743  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-lm 16959  df-haus 17043  df-1stc 17165
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