Theorem cncls2 23256

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 23232	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1127	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4536	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑌 → 𝑥 ⊆ 𝑌)
4	3	adantl 482	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ 𝑌)
5		toponuni 22897	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 733	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
7	4, 6	sseqtrd 3951	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ ∪ 𝐾)
8		eqid 2739	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	cncls2i 23253	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))
10	9	expcom 414	. . . . 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 3139	. . 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 517	. 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 23013	. . . . . . . . 9 ⊢ (Clsd‘𝐾) ⊆ 𝒫 ∪ 𝐾
15	5	ad2antlr 733	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑌 = ∪ 𝐾)
16	15	pweqd 4546	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝒫 𝑌 = 𝒫 ∪ 𝐾)
17	14, 16	sseqtrrid 3958	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (Clsd‘𝐾) ⊆ 𝒫 𝑌)
18	17	sseld 3914	. . . . . . 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 23025	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑥) = 𝑥)
21	20	ad2antll 735	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → ((cls‘𝐾)‘𝑥) = 𝑥)
22	21	imaeq2d 6012	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ ((cls‘𝐾)‘𝑥)) = (◡𝐹 “ 𝑥))
23	22	sseq2d 3947	. . . . . . . . 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 22896	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
25	24	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝐽 ∈ Top)
26		cnvimass 6034	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
27		fdm 6664	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
28	27	ad2antrl 734	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = 𝑋)
29		toponuni 22897	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	29	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → 𝑋 = ∪ 𝐽)
31	28, 30	eqtrd 2774	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → dom 𝐹 = ∪ 𝐽)
32	26, 31	sseqtrid 3957	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ 𝑥 ∈ (Clsd‘𝐾))) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
33		eqid 2739	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
34	33	iscld4 23048	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥)))
35	25, 32, 34	syl2anc 590	. . . . . . . . 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 283	. . . . . . . 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 457	. . . . . . 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 274	. . . . . 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 240	. . . . 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 3148	. . . 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 576	. . 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 23252	. . 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 260	. 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 213	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838  ^◡ccnv 5617  dom cdm 5618   “ cima 5621  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Topctop 22876  TopOnctopon 22893  Clsdccld 22999  clsccl 23001   Cn ccn 23207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-top 22877  df-topon 22894  df-cld 23002  df-cls 23004  df-cn 23210

This theorem is referenced by:  cncls  23257