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Theorem dnsconst 23102
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 7056). (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 765 . . 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 22970 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
5 ffn 6716 . . 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 1194 . . 3 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
8 simpll 763 . . . . . 6 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐾 ∈ Fre)
9 simpr1 1192 . . . . . 6 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑃𝑌)
103t1sncld 23050 . . . . . 6 ((𝐾 ∈ Fre ∧ 𝑃𝑌) → {𝑃} ∈ (Clsd‘𝐾))
118, 9, 10syl2anc 582 . . . . 5 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → {𝑃} ∈ (Clsd‘𝐾))
12 cnclima 22992 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {𝑃} ∈ (Clsd‘𝐾)) → (𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
131, 11, 12syl2anc 582 . . . 4 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
14 simpr2 1193 . . . 4 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ (𝐹 “ {𝑃}))
152clsss2 22796 . . . 4 (((𝐹 “ {𝑃}) ∈ (Clsd‘𝐽) ∧ 𝐴 ⊆ (𝐹 “ {𝑃})) → ((cls‘𝐽)‘𝐴) ⊆ (𝐹 “ {𝑃}))
1613, 14, 15syl2anc 582 . . 3 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) ⊆ (𝐹 “ {𝑃}))
177, 16eqsstrrd 4020 . 2 (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑋 ⊆ (𝐹 “ {𝑃}))
18 fconst3 7216 . 2 (𝐹:𝑋⟶{𝑃} ↔ (𝐹 Fn 𝑋𝑋 ⊆ (𝐹 “ {𝑃})))
196, 17, 18sylanbrc 581 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 394  w3a 1085   = wceq 1539  wcel 2104  wss 3947  {csn 4627   cuni 4907  ccnv 5674  cima 5678   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7411  Clsdccld 22740  clsccl 22742   Cn ccn 22948  Frect1 23031
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-top 22616  df-topon 22633  df-cld 22743  df-cls 22745  df-cn 22951  df-t1 23038
This theorem is referenced by:  ipasslem8  30357
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