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Theorem hausmapdom 19063
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 19520 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 19024 . . . . . . 7  |-  ( ( J  e.  1stc  /\  A  C_  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  E. f
( f : NN --> A  /\  f ( ~~> t `  J ) x ) ) )
323adant1 1001 . . . . . 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 6376 . . . . . . . . . . . 12  |-  ( J  e.  Haus  ->  U. J  e.  _V )
543ad2ant1 1004 . . . . . . . . . . 11  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  U. J  e. 
_V )
61, 5syl5eqel 2525 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  X  e.  _V )
7 simp3 985 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  C_  X
)
86, 7ssexd 4436 . . . . . . . . 9  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  A  e.  _V )
9 nnex 10324 . . . . . . . . 9  |-  NN  e.  _V
10 elmapg 7223 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  NN  e.  _V )  -> 
( f  e.  ( A  ^m  NN )  <-> 
f : NN --> A ) )
118, 9, 10sylancl 657 . . . . . . . 8  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( f  e.  ( A  ^m  NN ) 
<->  f : NN --> A ) )
1211anbi1d 699 . . . . . . 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 1685 . . . . . 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 256 . . . . 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 2719 . . . . 5  |-  ( E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x  <->  E. f
( f  e.  ( A  ^m  NN )  /\  f ( ~~> t `  J ) x ) )
1614, 15syl6bbr 263 . . . 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 2973 . . . . 5  |-  x  e. 
_V
1817elima 5171 . . . 4  |-  ( x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  <->  E. f  e.  ( A  ^m  NN ) f ( ~~> t `  J
) x )
1916, 18syl6bbr 263 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  ( ( ~~> t `  J ) " ( A  ^m  NN ) ) ) )
2019eqrdv 2439 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  =  ( ( ~~> t `  J
) " ( A  ^m  NN ) ) )
21 ovex 6115 . . 3  |-  ( A  ^m  NN )  e. 
_V
22 lmfun 18944 . . . 4  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
23223ad2ant1 1004 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  Fun  ( ~~> t `  J ) )
24 imadomg 8697 . . 3  |-  ( ( A  ^m  NN )  e.  _V  ->  ( Fun  ( ~~> t `  J
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) ) )
2521, 23, 24mpsyl 63 . 2  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( ~~> t `  J ) " ( A  ^m  NN ) )  ~<_  ( A  ^m  NN ) )
2620, 25eqbrtrd 4309 1  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761   E.wrex 2714   _Vcvv 2970    C_ wss 3325   U.cuni 4088   class class class wbr 4289   "cima 4839   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210    ~<_ cdom 7304   NNcn 10318   clsccl 18581   ~~> tclm 18789   Hauscha 18871   1stcc1stc 19000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-ac2 8628  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-acn 8108  df-ac 8282  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-top 18462  df-topon 18465  df-cld 18582  df-ntr 18583  df-cls 18584  df-lm 18792  df-haus 18878  df-1stc 19002
This theorem is referenced by:  hauspwdom  19064
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