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Theorem cnclima 22019
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)) → (𝐹𝐴) ∈ (Clsd‘𝐽))

Proof of Theorem cnclima
StepHypRef Expression
1 eqid 2738 . . . . . 6 𝐽 = 𝐽
2 eqid 2738 . . . . . 6 𝐾 = 𝐾
31, 2cnf 21997 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
43adantr 484 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → 𝐹: 𝐽 𝐾)
5 ffun 6507 . . . . . 6 (𝐹: 𝐽 𝐾 → Fun 𝐹)
6 funcnvcnv 6406 . . . . . 6 (Fun 𝐹 → Fun 𝐹)
7 imadif 6423 . . . . . 6 (Fun 𝐹 → (𝐹 “ ( 𝐾𝐴)) = ((𝐹 𝐾) ∖ (𝐹𝐴)))
85, 6, 73syl 18 . . . . 5 (𝐹: 𝐽 𝐾 → (𝐹 “ ( 𝐾𝐴)) = ((𝐹 𝐾) ∖ (𝐹𝐴)))
9 fimacnv 6526 . . . . . 6 (𝐹: 𝐽 𝐾 → (𝐹 𝐾) = 𝐽)
109difeq1d 4012 . . . . 5 (𝐹: 𝐽 𝐾 → ((𝐹 𝐾) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
118, 10eqtr2d 2774 . . . 4 (𝐹: 𝐽 𝐾 → ( 𝐽 ∖ (𝐹𝐴)) = (𝐹 “ ( 𝐾𝐴)))
124, 11syl 17 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ( 𝐽 ∖ (𝐹𝐴)) = (𝐹 “ ( 𝐾𝐴)))
132cldopn 21782 . . . 4 (𝐴 ∈ (Clsd‘𝐾) → ( 𝐾𝐴) ∈ 𝐾)
14 cnima 22016 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ( 𝐾𝐴) ∈ 𝐾) → (𝐹 “ ( 𝐾𝐴)) ∈ 𝐽)
1513, 14sylan2 596 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (𝐹 “ ( 𝐾𝐴)) ∈ 𝐽)
1612, 15eqeltrd 2833 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽)
17 cntop1 21991 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
18 cnvimass 5923 . . . 4 (𝐹𝐴) ⊆ dom 𝐹
1918, 4fssdm 6524 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (𝐹𝐴) ⊆ 𝐽)
201iscld2 21779 . . 3 ((𝐽 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐽) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
2117, 19, 20syl2an2r 685 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
2216, 21mpbird 260 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (𝐹𝐴) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  cdif 3840  wss 3843   cuni 4796  ccnv 5524  cima 5528  Fun wfun 6333  wf 6335  cfv 6339  (class class class)co 7170  Topctop 21644  Clsdccld 21767   Cn ccn 21975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-top 21645  df-topon 21662  df-cld 21770  df-cn 21978
This theorem is referenced by:  iscncl  22020  cncls2i  22021  paste  22045  cnt1  22101  dnsconst  22129  cnconn  22173  hauseqlcld  22397  txconn  22440  imasncld  22442  r0cld  22489  kqreglem2  22493  kqnrmlem1  22494  kqnrmlem2  22495  hmeocld  22518  nrmhmph  22545  tgphaus  22868  csscld  24001  clsocv  24002  hmeoclda  34160  hmeocldb  34161  rfcnpre3  42114  rfcnpre4  42115  sepfsepc  45743  seppcld  45745
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