Theorem cncls2 23251

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 23227	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1122	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4549	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑌 → 𝑥 ⊆ 𝑌)
4	3	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ 𝑌)
5		toponuni 22892	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 728	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
7	4, 6	sseqtrd 3959	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ ∪ 𝐾)
8		eqid 2737	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	cncls2i 23248	. . . . . 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 3138	. . 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 23008	. . . . . . . . 9 ⊢ (Clsd‘𝐾) ⊆ 𝒫 ∪ 𝐾
15	5	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑌 = ∪ 𝐾)
16	15	pweqd 4559	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝒫 𝑌 = 𝒫 ∪ 𝐾)
17	14, 16	sseqtrrid 3966	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (Clsd‘𝐾) ⊆ 𝒫 𝑌)
18	17	sseld 3921	. . . . . . 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 23020	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑥) = 𝑥)
21	20	ad2antll 730	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → ((cls‘𝐾)‘𝑥) = 𝑥)
22	21	imaeq2d 6020	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ ((cls‘𝐾)‘𝑥)) = (◡𝐹 “ 𝑥))
23	22	sseq2d 3955	. . . . . . . . 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 22891	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
25	24	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝐽 ∈ Top)
26		cnvimass 6042	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
27		fdm 6672	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
28	27	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝑋)
29		toponuni 22892	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	29	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝑋 = ∪ 𝐽)
31	28, 30	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = ∪ 𝐽)
32	26, 31	sseqtrid 3965	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
33		eqid 2737	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
34	33	iscld4 23043	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥)))
35	25, 32, 34	syl2anc 585	. . . . . . . . 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 282	. . . . . . . 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 273	. . . . . 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 239	. . . . 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 3147	. . . 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 23247	. . 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 259	. 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 212	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Topctop 22871  TopOnctopon 22888  Clsdccld 22994  clsccl 22996   Cn ccn 23202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-top 22872  df-topon 22889  df-cld 22997  df-cls 22999  df-cn 23205

This theorem is referenced by:  cncls  23252