Theorem cnclima 23191

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 2734	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2734	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23169	. . . . 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 6705	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
6		funcnvcnv 6599	. . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
7		imadif 6616	. . . . . 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 6724	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
10	9	difeq1d 4098	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → ((◡𝐹 “ ∪ 𝐾) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
11	8, 10	eqtr2d 2770	. . . 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 22954	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐾) → (∪ 𝐾 ∖ 𝐴) ∈ 𝐾)
14		cnima 23188	. . . 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 2833	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽)
17		cntop1 23163	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
18		cnvimass 6066	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
19	18, 4	fssdm 6721	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
20	1	iscld2 22951	. . 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 1539   ∈ wcel 2107   ∖ cdif 3921   ⊆ wss 3924  ∪ cuni 4880  ^◡ccnv 5650   “ cima 5654  Fun wfun 6521  ⟶wf 6523  ‘cfv 6527  (class class class)co 7399  Topctop 22816  Clsdccld 22939   Cn ccn 23147

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 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-map 8836  df-top 22817  df-topon 22834  df-cld 22942  df-cn 23150

This theorem is referenced by:  iscncl  23192  cncls2i  23193  paste  23217  cnt1  23273  dnsconst  23301  cnconn  23345  hauseqlcld  23569  txconn  23612  imasncld  23614  r0cld  23661  kqreglem2  23665  kqnrmlem1  23666  kqnrmlem2  23667  hmeocld  23690  nrmhmph  23717  tgphaus  24040  csscld  25186  clsocv  25187  hmeoclda  36272  hmeocldb  36273  rfcnpre3  44984  rfcnpre4  44985  sepfsepc  48781  seppcld  48783