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Theorem hausmapdom 17516
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 17973 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 17477 . . . . . . 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 4665 . . . . . . . . . . . 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 2488 . . . . . . . . . 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 4310 . . . . . . . . 9  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  e.  _V )
9 nnex 9962 . . . . . . . . 9  |-  NN  e.  _V
10 elmapg 6990 . . . . . . . . 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 2672 . . . . 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 2919 . . . . 5  |-  x  e. 
_V
1817elima 5167 . . . 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 2402 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  =  ( ( ~~> t `  J
) " ( A  ^m  NN ) ) )
21 ovex 6065 . . 3  |-  ( A  ^m  NN )  e. 
_V
22 lmfun 17399 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
23223ad2ant1 978 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  Fun  ( ~~> t `  J ) )
24 imadomg 8368 . . 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 4192 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 2667   _Vcvv 2916    C_ wss 3280   U.cuni 3975   class class class wbr 4172   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977    ~<_ cdom 7066   NNcn 9956   clsccl 17037   ~~> tclm 17244   Hauscha 17326   1stcc1stc 17453
This theorem is referenced by:  hauspwdom  17517
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-acn 7785  df-ac 7953  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-top 16918  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040  df-lm 17247  df-haus 17333  df-1stc 17455
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