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Theorem dnsconst 22083
 Description: If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that ((cls‘𝐽)‘𝐴) = 𝑋 means "𝐴 is dense in 𝑋 " and 𝐴 ⊆ (◡𝐹 “ {𝑃}) means "𝐹 is constant on 𝐴 " (see funconstss 6821). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
dnsconst.1 𝑋 = 𝐽
dnsconst.2 𝑌 = 𝐾
Assertion
Ref Expression
dnsconst (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})

Proof of Theorem dnsconst
StepHypRef Expression
1 simplr 768 . . 3 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹 ∈ (𝐽 Cn 𝐾))
2 dnsconst.1 . . . 4 𝑋 = 𝐽
3 dnsconst.2 . . . 4 𝑌 = 𝐾
42, 3cnf 21951 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
5 ffn 6502 . . 3 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
61, 4, 53syl 18 . 2 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹 Fn 𝑋)
7 simpr3 1193 . . 3 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
8 simpll 766 . . . . . 6 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐾 ∈ Fre)
9 simpr1 1191 . . . . . 6 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑃𝑌)
103t1sncld 22031 . . . . . 6 ((𝐾 ∈ Fre ∧ 𝑃𝑌) → {𝑃} ∈ (Clsd‘𝐾))
118, 9, 10syl2anc 587 . . . . 5 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → {𝑃} ∈ (Clsd‘𝐾))
12 cnclima 21973 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {𝑃} ∈ (Clsd‘𝐾)) → (𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
131, 11, 12syl2anc 587 . . . 4 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
14 simpr2 1192 . . . 4 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ (𝐹 “ {𝑃}))
152clsss2 21777 . . . 4 (((𝐹 “ {𝑃}) ∈ (Clsd‘𝐽) ∧ 𝐴 ⊆ (𝐹 “ {𝑃})) → ((cls‘𝐽)‘𝐴) ⊆ (𝐹 “ {𝑃}))
1613, 14, 15syl2anc 587 . . 3 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) ⊆ (𝐹 “ {𝑃}))
177, 16eqsstrrd 3933 . 2 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑋 ⊆ (𝐹 “ {𝑃}))
18 fconst3 6972 . 2 (𝐹:𝑋⟶{𝑃} ↔ (𝐹 Fn 𝑋𝑋 ⊆ (𝐹 “ {𝑃})))
196, 17, 18sylanbrc 586 1 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  {csn 4525  ∪ cuni 4801  ◡ccnv 5526   “ cima 5530   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155  Clsdccld 21721  clsccl 21723   Cn ccn 21929  Frect1 22012 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8423  df-top 21599  df-topon 21616  df-cld 21724  df-cls 21726  df-cn 21932  df-t1 22019 This theorem is referenced by:  ipasslem8  28724
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