Theorem cncls2 22332

Description: Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
cncls2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncls2

Step	Hyp	Ref	Expression
1		cnf2 22308	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1119	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4539	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑌 → 𝑥 ⊆ 𝑌)
4	3	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ 𝑌)
5		toponuni 21971	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 723	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
7	4, 6	sseqtrd 3957	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ ∪ 𝐾)
8		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	cncls2i 22329	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))
10	9	expcom 413	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐾 → (𝐹 ∈ (𝐽 Cn 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))))
11	7, 10	syl 17	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))))
12	11	ralrimdva 3112	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))))
13	2, 12	jcad 512	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))))
14	8	cldss2 22089	. . . . . . . . 9 ⊢ (Clsd‘𝐾) ⊆ 𝒫 ∪ 𝐾
15	5	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑌 = ∪ 𝐾)
16	15	pweqd 4549	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝒫 𝑌 = 𝒫 ∪ 𝐾)
17	14, 16	sseqtrrid 3970	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (Clsd‘𝐾) ⊆ 𝒫 𝑌)
18	17	sseld 3916	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (Clsd‘𝐾) → 𝑥 ∈ 𝒫 𝑌))
19	18	imim1d 82	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ 𝒫 𝑌 → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))))
20		cldcls 22101	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑥) = 𝑥)
21	20	ad2antll 725	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → ((cls‘𝐾)‘𝑥) = 𝑥)
22	21	imaeq2d 5958	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ ((cls‘𝐾)‘𝑥)) = (◡𝐹 “ 𝑥))
23	22	sseq2d 3949	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥)))
24		topontop 21970	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
25	24	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝐽 ∈ Top)
26		cnvimass 5978	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
27		fdm 6593	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
28	27	ad2antrl 724	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝑋)
29		toponuni 21971	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	29	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝑋 = ∪ 𝐽)
31	28, 30	eqtrd 2778	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = ∪ 𝐽)
32	26, 31	sseqtrid 3969	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
33		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
34	33	iscld4 22124	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥)))
35	25, 32, 34	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → ((◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥)))
36	23, 35	bitr4d 281	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽)))
37	36	expr 456	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (Clsd‘𝐾) → (((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)) ↔ (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))))
38	37	pm5.74d 272	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))) ↔ (𝑥 ∈ (Clsd‘𝐾) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))))
39	19, 38	sylibd 238	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ 𝒫 𝑌 → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝑥 ∈ (Clsd‘𝐾) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))))
40	39	ralimdv2 3101	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)) → ∀𝑥 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽)))
41	40	imdistanda 571	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))))
42		iscncl 22328	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))))
43	41, 42	sylibrd 258	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
44	13, 43	impbid 211	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ^◡ccnv 5579  dom cdm 5580   “ cima 5583  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Topctop 21950  TopOnctopon 21967  Clsdccld 22075  clsccl 22077   Cn ccn 22283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-top 21951  df-topon 21968  df-cld 22078  df-cls 22080  df-cn 22286

This theorem is referenced by:  cncls  22333