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Theorem cnconst 21309
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 6612 . . . 4 (𝐵𝑌 → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
21adantl 467 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
3 cnconst2 21308 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
433expa 1111 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
5 eleq1 2838 . . . 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 442 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 382   = wceq 1631  wcel 2145  {csn 4316   × cxp 5247  wf 6027  cfv 6031  (class class class)co 6793  TopOnctopon 20935   Cn ccn 21249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-topgen 16312  df-top 20919  df-topon 20936  df-cn 21252  df-cnp 21253
This theorem is referenced by:  xrge0mulc1cn  30327  cxpcncf2  40631
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