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Theorem hausmapdom 17443
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 17896 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 17404 . . . . . . 7  |-  ( ( J  e.  1stc  /\  A  C_  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  E. f
( f : NN --> A  /\  f ( ~~> t `  J ) x ) ) )
323adant1 974 . . . . . 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 958 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  C_  X
)
5 uniexg 4620 . . . . . . . . . . . 12  |-  ( J  e.  Haus  ->  U. J  e.  _V )
653ad2ant1 977 . . . . . . . . . . 11  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  U. J  e. 
_V )
71, 6syl5eqel 2450 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  X  e.  _V )
8 ssexg 4262 . . . . . . . . . 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 9899 . . . . . . . . 9  |-  NN  e.  _V
11 elmapg 6928 . . . . . . . . 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 1631 . . . . . 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 2634 . . . . 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 2876 . . . . 5  |-  x  e. 
_V
1918elima 5120 . . . 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 2364 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  =  ( ( ~~> t `  J
) " ( A  ^m  NN ) ) )
22 ovex 6006 . . 3  |-  ( A  ^m  NN )  e. 
_V
23 lmfun 17326 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
24233ad2ant1 977 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  Fun  ( ~~> t `  J ) )
25 imadomg 8306 . . 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 4145 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 935   E.wex 1546    = wceq 1647    e. wcel 1715   E.wrex 2629   _Vcvv 2873    C_ wss 3238   U.cuni 3929   class class class wbr 4125   "cima 4795   Fun wfun 5352   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915    ~<_ cdom 7004   NNcn 9893   clsccl 16972   ~~> tclm 17173   Hauscha 17253   1stcc1stc 17380
This theorem is referenced by:  hauspwdom  17444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cc 8208  ax-ac2 8236  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719  df-acn 7722  df-ac 7890  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-top 16853  df-topon 16856  df-cld 16973  df-ntr 16974  df-cls 16975  df-lm 17176  df-haus 17260  df-1stc 17382
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