Theorem iscncl 23278

Description: A characterization of a continuity function using closed sets. Theorem 1(d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
iscncl	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))

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

Proof of Theorem iscncl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnf2 23258	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1118	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
3		cnclima 23277	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
4	3	ralrimiva 3145	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
5	4	adantl 481	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
6	2, 5	jca 511	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽)))
7		simprl 770	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋⟶𝑌)
8		toponuni 22921	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
9	8	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐽)
10		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋⟶𝑌)
11		fimacnv 6757	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
12	11	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → 𝑋 = (◡𝐹 “ 𝑌))
13	10, 12	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝑋 = (◡𝐹 “ 𝑌))
14	9, 13	eqtr3d 2778	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → ∪ 𝐽 = (◡𝐹 “ 𝑌))
15	14	difeq1d 4124	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑥)))
16		ffun 6738	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
17		funcnvcnv 6632	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
18		imadif 6649	. . . . . . . 8 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝑥)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑥)))
19	10, 16, 17, 18	4syl 19	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ (𝑌 ∖ 𝑥)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑥)))
20	15, 19	eqtr4d 2779	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) = (◡𝐹 “ (𝑌 ∖ 𝑥)))
21		imaeq2 6073	. . . . . . . 8 ⊢ (𝑦 = (𝑌 ∖ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝑌 ∖ 𝑥)))
22	21	eleq1d 2825	. . . . . . 7 ⊢ (𝑦 = (𝑌 ∖ 𝑥) → ((◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽) ↔ (◡𝐹 “ (𝑌 ∖ 𝑥)) ∈ (Clsd‘𝐽)))
23		simplrr 777	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
24		toponuni 22921	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
25	24	ad3antlr 731	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝑌 = ∪ 𝐾)
26	25	difeq1d 4124	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (𝑌 ∖ 𝑥) = (∪ 𝐾 ∖ 𝑥))
27		topontop 22920	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
28	27	ad3antlr 731	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ Top)
29		eqid 2736	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
30	29	opncld 23042	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑥 ∈ 𝐾) → (∪ 𝐾 ∖ 𝑥) ∈ (Clsd‘𝐾))
31	28, 30	sylancom 588	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (∪ 𝐾 ∖ 𝑥) ∈ (Clsd‘𝐾))
32	26, 31	eqeltrd 2840	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (𝑌 ∖ 𝑥) ∈ (Clsd‘𝐾))
33	22, 23, 32	rspcdva 3622	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ (𝑌 ∖ 𝑥)) ∈ (Clsd‘𝐽))
34	20, 33	eqeltrd 2840	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽))
35		topontop 22920	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
36	35	ad3antrrr 730	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top)
37		cnvimass 6099	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
38	37, 10	fssdm 6754	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
39	38, 9	sseqtrd 4019	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
40		eqid 2736	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	isopn2 23041	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽)))
42	36, 39, 41	syl2anc 584	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽)))
43	34, 42	mpbird 257	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
44	43	ralrimiva 3145	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)
45		iscn 23244	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
46	45	adantr 480	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
47	7, 44, 46	mpbir2and 713	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → 𝐹 ∈ (𝐽 Cn 𝐾))
48	6, 47	impbida 800	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ∖ cdif 3947   ⊆ wss 3950  ∪ cuni 4906  ^◡ccnv 5683   “ cima 5687  Fun wfun 6554  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  Topctop 22900  TopOnctopon 22917  Clsdccld 23025   Cn ccn 23233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-top 22901  df-topon 22918  df-cld 23028  df-cn 23236

This theorem is referenced by:  cncls2  23282  paste  23303  cmphaushmeo  23809  ubthlem1  30890  ubthlem2  30891  rhmpreimacn  33885