Theorem cnclima 13017

Description: A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnclima	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))

Proof of Theorem cnclima

Step	Hyp	Ref	Expression
1		eqid 2170	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2170	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 12998	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantr 274	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5		ffun 5350	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
6		funcnvcnv 5257	. . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
7		imadif 5278	. . . . . 6 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)))
8	5, 6, 7	3syl 17	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)))
9		fimacnv 5625	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
10	9	difeq1d 3244	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
11	8, 10	eqtr2d 2204	. . . 4 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)))
12	4, 11	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)))
13	2	cldopn 12901	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐾) → (∪ 𝐾 ∖ 𝐴) ∈ 𝐾)
14		cnima 13014	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽)
15	13, 14	sylan2 284	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽)
16	12, 15	eqeltrd 2247	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)
17		cntop1 12995	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
18	17	adantr 274	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐽 ∈ Top)
19		cnvimass 4974	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
20	19, 4	fssdm 5362	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
21	1	iscld2 12898	. . 3 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
22	18, 20, 21	syl2anc 409	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
23	16, 22	mpbird 166	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141   ∖ cdif 3118   ⊆ wss 3121  ∪ cuni 3796  ^◡ccnv 4610   “ cima 4614  Fun wfun 5192  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  Topctop 12789  Clsdccld 12886   Cn ccn 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cld 12889  df-cn 12982

This theorem is referenced by:  hmeocld  13106