seppcld - Mathbox for Zhi Wang

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Theorem seppcld 48060

Description: If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)

**Hypothesis**
Ref	Expression
seppsepf.1	⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1})))

**Assertion**
Ref	Expression
seppcld	⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))

Distinct variable groups: 𝑓,𝐽 𝑆,𝑓 𝑇,𝑓 Allowed substitution hint: 𝜑(𝑓)

**Proof of Theorem seppcld**
Step	Hyp	Ref	Expression
1		seppsepf.1	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1})))
2		simprl 769	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → 𝑆 = (^◡𝑓 “ {0}))
3		simpl 481	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → 𝑓 ∈ (𝐽 Cn II))
4		0xr 11291	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
5		iccid 13401	. . . . . . . 8 ⊢ (0 ∈ ℝ^* → (0[,]0) = {0})
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ (0[,]0) = {0}
7		0le0 12343	. . . . . . . 8 ⊢ 0 ≤ 0
8		0le1 11767	. . . . . . . 8 ⊢ 0 ≤ 1
9		icccldii 48049	. . . . . . . 8 ⊢ ((0 ≤ 0 ∧ 0 ≤ 1) → (0[,]0) ∈ (Clsd‘II))
10	7, 8, 9	mp2an 690	. . . . . . 7 ⊢ (0[,]0) ∈ (Clsd‘II)
11	6, 10	eqeltrri 2822	. . . . . 6 ⊢ {0} ∈ (Clsd‘II)
12		cnclima 23190	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {0} ∈ (Clsd‘II)) → (^◡𝑓 “ {0}) ∈ (Clsd‘𝐽))
13	3, 11, 12	sylancl 584	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → (^◡𝑓 “ {0}) ∈ (Clsd‘𝐽))
14	2, 13	eqeltrd 2825	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → 𝑆 ∈ (Clsd‘𝐽))
15		simprr 771	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → 𝑇 = (^◡𝑓 “ {1}))
16		1xr 11303	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
17		iccid 13401	. . . . . . . 8 ⊢ (1 ∈ ℝ^* → (1[,]1) = {1})
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (1[,]1) = {1}
19		1le1 11872	. . . . . . . 8 ⊢ 1 ≤ 1
20		icccldii 48049	. . . . . . . 8 ⊢ ((0 ≤ 1 ∧ 1 ≤ 1) → (1[,]1) ∈ (Clsd‘II))
21	8, 19, 20	mp2an 690	. . . . . . 7 ⊢ (1[,]1) ∈ (Clsd‘II)
22	18, 21	eqeltrri 2822	. . . . . 6 ⊢ {1} ∈ (Clsd‘II)
23		cnclima 23190	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {1} ∈ (Clsd‘II)) → (^◡𝑓 “ {1}) ∈ (Clsd‘𝐽))
24	3, 22, 23	sylancl 584	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → (^◡𝑓 “ {1}) ∈ (Clsd‘𝐽))
25	15, 24	eqeltrd 2825	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → 𝑇 ∈ (Clsd‘𝐽))
26	14, 25	jca 510	. . 3 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1}))) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
27	26	rexlimiva 3137	. 2 ⊢ (∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (^◡𝑓 “ {0}) ∧ 𝑇 = (^◡𝑓 “ {1})) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
28	1, 27	syl 17	1 ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 {csn 4624 class class class wbr 5143 ^◡ccnv 5671 “ cima 5675 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 ℝ^*cxr 11277 ≤ cle 11279 [,]cicc 13359 Clsdccld 22938 Cn ccn 23146 IIcii 24813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-rest 17403 df-topgen 17424 df-ordt 17482 df-ps 18557 df-tsr 18558 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-top 22814 df-topon 22831 df-bases 22867 df-cld 22941 df-cn 23149 df-ii 24815

This theorem is referenced by: (None)

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