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Theorem cnconst 20993
Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵𝑌𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem cnconst
StepHypRef Expression
1 fconst2g 6423 . . . 4 (𝐵𝑌 → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
21adantl 482 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
3 cnconst2 20992 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
433expa 1262 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
5 eleq1 2692 . . . 4 (𝐹 = (𝑋 × {𝐵}) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾)))
64, 5syl5ibrcom 237 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝐹 = (𝑋 × {𝐵}) → 𝐹 ∈ (𝐽 Cn 𝐾)))
72, 6sylbid 230 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝐹:𝑋⟶{𝐵} → 𝐹 ∈ (𝐽 Cn 𝐾)))
87impr 648 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵𝑌𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  {csn 4153   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  TopOnctopon 20613   Cn ccn 20933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-topgen 16020  df-top 20616  df-topon 20618  df-cn 20936  df-cnp 20937
This theorem is referenced by:  xrge0mulc1cn  29761  cxpcncf2  39404
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