met1stc - Metamath Proof Explorer

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Theorem met1stc 22326

Description: The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

**Hypothesis**
Ref	Expression
methaus.1

**Assertion**
Ref	Expression
met1stc

Proof of Theorem met1stc Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		methaus.1	. . 3
2	1	mopntop 22245	. 2
3	1	mopnuni 22246	. . . . . 6
4	3	eleq2d 2687	. . . . 5
5	4	biimpar 502	. . . 4 $/\$
6		simpll 790	. . . . . . . . 9 $/\$ $/\$
7		simplr 792	. . . . . . . . 9 $/\$ $/\$
8		nnrp 11842	. . . . . . . . . . . 12
9	8	adantl 482	. . . . . . . . . . 11 $/\$ $/\$
10	9	rpreccld 11882	. . . . . . . . . 10 $/\$ $/\$
11	10	rpxrd 11873	. . . . . . . . 9 $/\$ $/\$
12	1	blopn 22305	. . . . . . . . 9 $/\$ $/\$
13	6, 7, 11, 12	syl3anc 1326	. . . . . . . 8 $/\$ $/\$
14		eqid 2622	. . . . . . . 8
15	13, 14	fmptd 6385	. . . . . . 7 $/\$
16		frn 6053	. . . . . . 7
17	15, 16	syl 17	. . . . . 6 $/\$
18		nnex 11026	. . . . . . . . 9
19	18	mptex 6486	. . . . . . . 8
20	19	rnex 7100	. . . . . . 7
21	20	elpw 4164	. . . . . 6
22	17, 21	sylibr 224	. . . . 5 $/\$
23		omelon 8543	. . . . . . . . 9
24		nnenom 12779	. . . . . . . . . 10
25	24	ensymi 8006	. . . . . . . . 9
26		isnumi 8772	. . . . . . . . 9 $/\$
27	23, 25, 26	mp2an 708	. . . . . . . 8
28		ovex 6678	. . . . . . . . . 10
29	28, 14	fnmpti 6022	. . . . . . . . 9
30		dffn4 6121	. . . . . . . . 9
31	29, 30	mpbi 220	. . . . . . . 8
32		fodomnum 8880	. . . . . . . 8
33	27, 31, 32	mp2 9	. . . . . . 7
34		domentr 8015	. . . . . . 7 $/\$
35	33, 24, 34	mp2an 708	. . . . . 6
36	35	a1i 11	. . . . 5 $/\$
37		simpll 790	. . . . . . . . . 10 $/\$ $/\$ $/\$
38		simprl 794	. . . . . . . . . 10 $/\$ $/\$ $/\$
39		simprr 796	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	1	mopni2 22298	. . . . . . . . . 10 $/\$ $/\$
41	37, 38, 39, 40	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
42		simp-4l 806	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simp-4r 807	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		simprl 794	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	44	nnrpd 11870	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	45	rpreccld 11882	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		blcntr 22218	. . . . . . . . . . . 12 $/\$ $/\$
48	42, 43, 46, 47	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	46	rpxrd 11873	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		simplrl 800	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	50	rpxrd 11873	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		nnrecre 11057	. . . . . . . . . . . . . . 15
53	52	ad2antrl 764	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	50	rpred 11872	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55		simprr 796	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	53, 54, 55	ltled 10185	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		ssbl 22228	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
58	42, 43, 49, 51, 56, 57	syl221anc 1337	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59		simplrr 801	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	58, 59	sstrd 3613	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	48, 60	jca 554	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62		elrp 11834	. . . . . . . . . . . 12 $/\$
63		nnrecl 11290	. . . . . . . . . . . 12 $/\$
64	62, 63	sylbi 207	. . . . . . . . . . 11
65	64	ad2antrl 764	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
66	61, 65	reximddv 3018	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	41, 66	rexlimddv 3035	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
68		ovexd 6680	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
69		vex 3203	. . . . . . . . . 10
70		oveq2 6658	. . . . . . . . . . . . 13
71	70	oveq2d 6666	. . . . . . . . . . . 12
72	71	cbvmptv 4750	. . . . . . . . . . 11
73	72	elrnmpt 5372	. . . . . . . . . 10
74	69, 73	mp1i 13	. . . . . . . . 9 $/\$ $/\$ $/\$
75		eleq2 2690	. . . . . . . . . . 11
76		sseq1 3626	. . . . . . . . . . 11
77	75, 76	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$
78	77	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	68, 74, 78	rexxfr2d 4883	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
80	67, 79	mpbird 247	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
81	80	expr 643	. . . . . 6 $/\$ $/\$ $/\$
82	81	ralrimiva 2966	. . . . 5 $/\$ $/\$
83		breq1 4656	. . . . . . 7
84		rexeq 3139	. . . . . . . . 9 $/\$ $/\$
85	84	imbi2d 330	. . . . . . . 8 $/\$ $/\$
86	85	ralbidv 2986	. . . . . . 7 $/\$ $/\$
87	83, 86	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $/\$
88	87	rspcev 3309	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
89	22, 36, 82, 88	syl12anc 1324	. . . 4 $/\$ $/\$ $/\$
90	5, 89	syldan 487	. . 3 $/\$ $/\$ $/\$
91	90	ralrimiva 2966	. 2 $/\$ $/\$
92		eqid 2622	. . 3
93	92	is1stc2 21245	. 2 $/\$ $/\$ $/\$
94	2, 91, 93	sylanbrc 698	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200

wss 3574

cpw 4158

cuni 4436 class class class wbr 4653

cmpt 4729

cdm 5114

crn 5115

con0 5723

wfn 5883

wf 5884

wfo 5886

cfv 5888 (class class class)co 6650

com 7065

cen 7952

cdom 7953

ccrd 8761

cr 9935

cc0 9936

c1 9937

cxr 10073

clt 10074

cle 10075

cdiv 10684

cn 11020

crp 11832

cxmt 19731

cbl 19733

cmopn 19736

ctop 20698

c1stc 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-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 ax-pre-sup 10014

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-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-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-1stc 21242

This theorem is referenced by: metelcls 23103 metcnp4 23108 metcn4 23109

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