Theorem iscncl 23478

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 23458	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1136	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
3		cnclima 23477	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
4	3	ralrimiva 3159	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
5	4	adantl 487	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
6	2, 5	jca 521	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽)))
7		simprl 783	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋⟶𝑌)
8		toponuni 23123	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
9	8	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐽)
10		simplrl 789	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋⟶𝑌)
11		fimacnv 6732	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
12	11	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → 𝑋 = (◡𝐹 “ 𝑌))
13	10, 12	syl 18	. . . . . . . . 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 2802	. . . . . . . 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 4080	. . . . . . 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 6712	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
17		funcnvcnv 6607	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
18		imadif 6624	. . . . . . . 8 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝑥)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑥)))
19	10, 16, 17, 18	4syl 20	. . . . . . 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 2803	. . . . . 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 6060	. . . . . . . 8 ⊢ (𝑦 = (𝑌 ∖ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝑌 ∖ 𝑥)))
22	21	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = (𝑌 ∖ 𝑥) → ((◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽) ↔ (◡𝐹 “ (𝑌 ∖ 𝑥)) ∈ (Clsd‘𝐽)))
23		simplrr 790	. . . . . . 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 23123	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
25	24	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝑌 = ∪ 𝐾)
26	25	difeq1d 4080	. . . . . . . 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 23122	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
28	27	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ Top)
29		eqid 2765	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
30	29	opncld 23242	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑥 ∈ 𝐾) → (∪ 𝐾 ∖ 𝑥) ∈ (Clsd‘𝐾))
31	28, 30	sylancom 600	. . . . . . . 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 2865	. . . . . . 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 3584	. . . . . 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 2865	. . . . 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 23122	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
36	35	ad3antrrr 743	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top)
37		cnvimass 6086	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
38	37, 10	fssdm 6729	. . . . . . 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 3974	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
40		eqid 2765	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	isopn2 23241	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽)))
42	36, 39, 41	syl2anc 596	. . . . 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 260	. . . 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 3159	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)
45		iscn 23444	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
46	45	adantr 486	. . 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 726	. 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 813	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874  ^◡ccnv 5662   “ cima 5666  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Topctop 23102  TopOnctopon 23119  Clsdccld 23225   Cn ccn 23433

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-top 23103  df-topon 23120  df-cld 23228  df-cn 23436

This theorem is used by:  cncls2  23482  paste  23503  cmphaushmeo  24010  ubthlem1  31295  ubthlem2  31296  rhmpreimacn  34341