1stccnp - Metamath Proof Explorer

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Theorem 1stccnp 21265

Description: A mapping is continuous at

in a first-countable space

iff it is sequentially continuous at

, meaning that the image under

of every sequence converging at

converges to

. This proof uses ax-cc 9257, but only via 1stcelcls 21264, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.)

**Hypotheses**
Ref	Expression
1stccnp.1
1stccnp.2	TopOn
1stccnp.3	TopOn
1stccnp.4

**Assertion**
Ref	Expression
1stccnp	$/\$ $/\$

Distinct variable groups:

Proof of Theorem 1stccnp Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stccnp.2	. . . . 5 TopOn
2		1stccnp.3	. . . . 5 TopOn
3	1, 2	jca 554	. . . 4 TopOn $/\$ TopOn
4		cnpf2 21054	. . . . 5 TopOn $/\$ TopOn $/\$
5	4	3expa 1265	. . . 4 TopOn $/\$ TopOn $/\$
6	3, 5	sylan 488	. . 3 $/\$
7		simprr 796	. . . . . 6 $/\$ $/\$ $/\$
8		simplr 792	. . . . . 6 $/\$ $/\$ $/\$
9	7, 8	lmcnp 21108	. . . . 5 $/\$ $/\$ $/\$
10	9	ex 450	. . . 4 $/\$ $/\$
11	10	alrimiv 1855	. . 3 $/\$ $/\$
12	6, 11	jca 554	. 2 $/\$ $/\$ $/\$
13		simprl 794	. . 3 $/\$ $/\$ $/\$
14		fal 1490	. . . . . . . . 9
15		19.29 1801	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$ $/\$
16		simprl 794	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$
17		difss 3737	. . . . . . . . . . . . . . . . . . . . 21 $\$
18		fss 6056	. . . . . . . . . . . . . . . . . . . . 21 $\$ $/\$ $\$
19	16, 17, 18	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $\$ $/\$
20		simprr 796	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $\$ $/\$
21	19, 20	jca 554	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
22		nnuz 11723	. . . . . . . . . . . . . . . . . . . . . 22
23		simplrr 801	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
24		1zzd 11408	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
25		simprr 796	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
26		simplrl 800	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
27	22, 23, 24, 25, 26	lmcvg 21066	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
28	22	r19.2uz 14091	. . . . . . . . . . . . . . . . . . . . . 22
29		simprll 802	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
30		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 $\$
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
32		fvco2 6273	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$
33	31, 32	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
34	33	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
35	29	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
36	35	eldifad 3586	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
37		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 $/\$ $/\$ $/\$
38	37	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
39		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . . . 27
40		elpreima 6337	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 $/\$
41	38, 39, 40	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $/\$
42	35	eldifbd 3587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
43	42	pm2.21d 118	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
44	41, 43	sylbird 250	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $/\$
45	36, 44	mpand 711	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
46	34, 45	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
47	46	rexlimdva 3031	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
48	28, 47	syl5 34	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
49	27, 48	mpd 15	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
50	49	expr 643	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $\$ $/\$
51	21, 50	embantd 59	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
52	51	ex 450	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $\$ $/\$ $/\$
53	52	com23 86	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $\$ $/\$
54	53	impd 447	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
55	54	exlimdv 1861	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
56	15, 55	syl5 34	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
57	56	exp4b 632	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$
58	57	com23 86	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$
59	58	impr 649	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\$ $/\$
60	59	imp 445	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
61	14, 60	mtoi 190	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
62		1stccnp.1	. . . . . . . . . 10
63	62	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
64	1	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ TopOn
65		toponuni 20719	. . . . . . . . . . 11 TopOn
66	64, 65	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
67	17, 66	syl5sseq 3653	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
68		eqid 2622	. . . . . . . . . 10
69	68	1stcelcls 21264	. . . . . . . . 9 $/\$ $\$ $\$ $\$ $/\$
70	63, 67, 69	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $\$ $/\$
71	61, 70	mtbird 315	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
72		topontop 20718	. . . . . . . . 9 TopOn
73	64, 72	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
74		1stccnp.4	. . . . . . . . . 10
75	74	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 66	eleqtrd 2703	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
77	68	elcls 20877	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $\$
78	73, 67, 76, 77	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $\$
79	71, 78	mtbid 314	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
80	13	ad2antrr 762	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81		ffun 6048	. . . . . . . . . . . . 13
82	80, 81	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		toponss 20731	. . . . . . . . . . . . . 14 TopOn $/\$
84	64, 83	sylan 488	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		fdm 6051	. . . . . . . . . . . . . 14
86	80, 85	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	84, 86	sseqtr4d 3642	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88		funimass3 6333	. . . . . . . . . . . 12 $/\$
89	82, 87, 88	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90		df-ss 3588	. . . . . . . . . . . . 13
91	84, 90	sylib 208	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	91	sseq1d 3632	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
93	89, 92	bitr4d 271	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
94		nne 2798	. . . . . . . . . . 11 $\$ $\$
95		inssdif0 3947	. . . . . . . . . . 11 $\$
96	94, 95	bitr4i 267	. . . . . . . . . 10 $\$
97	93, 96	syl6bbr 278	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\$
98	97	anbi2d 740	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\$
99	98	rexbidva 3049	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\$
100		rexanali 2998	. . . . . . 7 $/\$ $\$ $\$
101	99, 100	syl6bb 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\$
102	79, 101	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	102	expr 643	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ralrimiva 2966	. . 3 $/\$ $/\$ $/\$ $/\$
105		iscnp 21041	. . . . 5 TopOn $/\$ TopOn $/\$ $/\$ $/\$
106	1, 2, 74, 105	syl3anc 1326	. . . 4 $/\$ $/\$
107	106	adantr 481	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
108	13, 104, 107	mpbir2and 957	. 2 $/\$ $/\$ $/\$
109	12, 108	impbida 877	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wal 1481

wceq 1483

wfal 1488

wex 1704

wcel 1990

wne 2794

wral 2912

wrex 2913 $\$ cdif 3571

cin 3573

wss 3574

c0 3915

cuni 4436 class class class wbr 4653

ccnv 5113

cdm 5114

cima 5117

ccom 5118

wfun 5882

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

c1 9937

cn 11020

cuz 11687

ctop 20698 TopOnctopon 20715

ccl 20822

ccnp 21029

clm 21030

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-cc 9257 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-fal 1489 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-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-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-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-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-cnp 21032 df-lm 21033 df-1stc 21242

This theorem is referenced by: 1stccn 21266 metcnp4 23108

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