Theorem hausmapdom 21303

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 21792 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.

Step	Hyp	Ref	Expression
1		hauspwdom.1	. . . . . . . 8 |- X = U. J
2	1	1stcelcls 21264	. . . . . . 7 |- ( ( J e. 1stc /\ A C_ X ) -> ( x e. ( ( cls ` J ) ` A ) <-> E. f ( f : NN --> A /\ f ( ~~> t ` J ) x ) ) )
3	2	3adant1 1079	. . . . . 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 6955	. . . . . . . . . . . 12 |- ( J e. Haus -> U. J e. _V )
5	4	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> U. J e. _V )
6	1, 5	syl5eqel 2705	. . . . . . . . . 10 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> X e. _V )
7		simp3 1063	. . . . . . . . . 10 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> A C_ X )
8	6, 7	ssexd 4805	. . . . . . . . 9 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> A e. _V )
9		nnex 11026	. . . . . . . . 9 |- NN e. _V
10		elmapg 7870	. . . . . . . . 9 |- ( ( A e. _V /\ NN e. _V ) -> ( f e. ( A ^m NN ) <-> f : NN --> A ) )
11	8, 9, 10	sylancl 694	. . . . . . . 8 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( f e. ( A ^m NN ) <-> f : NN --> A ) )
12	11	anbi1d 741	. . . . . . 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 ) ) )
13	12	exbidv 1850	. . . . . 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 ) ) )
14	3, 13	bitr4d 271	. . . . 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 2918	. . . . 5 |- ( E. f e. ( A ^m NN ) f ( ~~> t ` J ) x <-> E. f ( f e. ( A ^m NN ) /\ f ( ~~> t ` J ) x ) )
16	14, 15	syl6bbr 278	. . . 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 3203	. . . . 5 |- x e. _V
18	17	elima 5471	. . . 4 |- ( x e. ( ( ~~> t ` J ) " ( A ^m NN ) ) <-> E. f e. ( A ^m NN ) f ( ~~> t ` J ) x )
19	16, 18	syl6bbr 278	. . 3 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( x e. ( ( cls ` J ) ` A ) <-> x e. ( ( ~~> t ` J ) " ( A ^m NN ) ) ) )
20	19	eqrdv 2620	. 2 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( ( cls ` J ) ` A ) = ( ( ~~> t ` J ) " ( A ^m NN ) ) )
21		ovex 6678	. . 3 |- ( A ^m NN ) e. _V
22		lmfun 21185	. . . 4 |- ( J e. Haus -> Fun ( ~~> t ` J ) )
23	22	3ad2ant1 1082	. . 3 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> Fun ( ~~> t ` J ) )
24		imadomg 9356	. . 3 |- ( ( A ^m NN ) e. _V -> ( Fun ( ~~> t ` J ) -> ( ( ~~> t ` J ) " ( A ^m NN ) ) ~<_ ( A ^m NN ) ) )
25	21, 23, 24	mpsyl 68	. 2 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( ( ~~> t ` J ) " ( A ^m NN ) ) ~<_ ( A ^m NN ) )
26	20, 25	eqbrtrd 4675	1 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( ( cls ` J ) ` A ) ~<_ ( A ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ~<_ cdom 7953   NNcn 11020   clsccl 20822   ~~> tclm 21030   Hauscha 21112   1stcc1stc 21240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033  df-haus 21119  df-1stc 21242

This theorem is referenced by:  hauspwdom  21304