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Theorem hausmapdom 17524
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 17981 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 17485 . . . . . . 7  |-  ( ( J  e.  1stc  /\  A  C_  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  E. f
( f : NN --> A  /\  f ( ~~> t `  J ) x ) ) )
323adant1 975 . . . . . 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 uniexg 4673 . . . . . . . . . . . 12  |-  ( J  e.  Haus  ->  U. J  e.  _V )
543ad2ant1 978 . . . . . . . . . . 11  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  U. J  e. 
_V )
61, 5syl5eqel 2496 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  X  e.  _V )
7 simp3 959 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  C_  X
)
86, 7ssexd 4318 . . . . . . . . 9  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  e.  _V )
9 nnex 9970 . . . . . . . . 9  |-  NN  e.  _V
10 elmapg 6998 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  NN  e.  _V )  -> 
( f  e.  ( A  ^m  NN )  <-> 
f : NN --> A ) )
118, 9, 10sylancl 644 . . . . . . . 8  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( f  e.  ( A  ^m  NN ) 
<->  f : NN --> A ) )
1211anbi1d 686 . . . . . . 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 ) ) )
1312exbidv 1633 . . . . . 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 ) ) )
143, 13bitr4d 248 . . . . 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 ) ) )
15 df-rex 2680 . . . . 5  |-  ( E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x  <->  E. f
( f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x ) )
1614, 15syl6bbr 255 . . . 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 ) )
17 vex 2927 . . . . 5  |-  x  e. 
_V
1817elima 5175 . . . 4  |-  ( x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  <->  E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x )
1916, 18syl6bbr 255 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) ) ) )
2019eqrdv 2410 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  =  ( ( ~~> t `  J
) " ( A  ^m  NN ) ) )
21 ovex 6073 . . 3  |-  ( A  ^m  NN )  e. 
_V
22 lmfun 17407 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
23223ad2ant1 978 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  Fun  ( ~~> t `  J ) )
24 imadomg 8376 . . 3  |-  ( ( A  ^m  NN )  e.  _V  ->  ( Fun  ( ~~> t `  J
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) ) )
2521, 23, 24mpsyl 61 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) )
2620, 25eqbrtrd 4200 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924    C_ wss 3288   U.cuni 3983   class class class wbr 4180   "cima 4848   Fun wfun 5415   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^m cmap 6985    ~<_ cdom 7074   NNcn 9964   clsccl 17045   ~~> tclm 17252   Hauscha 17334   1stcc1stc 17461
This theorem is referenced by:  hauspwdom  17525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cc 8279  ax-ac2 8307  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-acn 7793  df-ac 7961  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-top 16926  df-topon 16929  df-cld 17046  df-ntr 17047  df-cls 17048  df-lm 17255  df-haus 17341  df-1stc 17463
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