Theorem cnclima 23155

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 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2729	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23133	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4	3	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5		ffun 6691	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
6		funcnvcnv 6583	. . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
7		imadif 6600	. . . . . 6 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)))
8	5, 6, 7	3syl 18	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) = ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)))
9		fimacnv 6710	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
10	9	difeq1d 4088	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
11	8, 10	eqtr2d 2765	. . . 4 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)))
12	4, 11	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) = (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)))
13	2	cldopn 22918	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐾) → (∪ 𝐾 ∖ 𝐴) ∈ 𝐾)
14		cnima 23152	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (∪ 𝐾 ∖ 𝐴) ∈ 𝐾) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽)
15	13, 14	sylan2 593	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ (∪ 𝐾 ∖ 𝐴)) ∈ 𝐽)
16	12, 15	eqeltrd 2828	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)
17		cntop1 23127	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
18		cnvimass 6053	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
19	18, 4	fssdm 6707	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
20	1	iscld2 22915	. . 3 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
21	17, 19, 20	syl2an2r 685	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
22	16, 21	mpbird 257	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911   ⊆ wss 3914  ∪ cuni 4871  ^◡ccnv 5637   “ cima 5641  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Topctop 22780  Clsdccld 22903   Cn ccn 23111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-top 22781  df-topon 22798  df-cld 22906  df-cn 23114

This theorem is referenced by:  iscncl  23156  cncls2i  23157  paste  23181  cnt1  23237  dnsconst  23265  cnconn  23309  hauseqlcld  23533  txconn  23576  imasncld  23578  r0cld  23625  kqreglem2  23629  kqnrmlem1  23630  kqnrmlem2  23631  hmeocld  23654  nrmhmph  23681  tgphaus  24004  csscld  25149  clsocv  25150  hmeoclda  36321  hmeocldb  36322  rfcnpre3  45027  rfcnpre4  45028  sepfsepc  48916  seppcld  48918