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Theorem htpycn 23180
 Description: A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
ishtpy.3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
Assertion
Ref Expression
htpycn (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))

Proof of Theorem htpycn
Dummy variables 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishtpy.1 . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 ishtpy.3 . . . 4 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
3 ishtpy.4 . . . 4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
41, 2, 3ishtpy 23179 . . 3 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ ( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)))))
5 simpl 476 . . 3 (( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))) → ∈ ((𝐽 ×t II) Cn 𝐾))
64, 5syl6bi 245 . 2 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) → ∈ ((𝐽 ×t II) Cn 𝐾)))
76ssrdv 3826 1 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∀wral 3089   ⊆ wss 3791  ‘cfv 6135  (class class class)co 6922  0cc0 10272  1c1 10273  TopOnctopon 21122   Cn ccn 21436   ×t ctx 21772  IIcii 23086   Htpy chtpy 23174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-top 21106  df-topon 21123  df-cn 21439  df-htpy 23177 This theorem is referenced by:  htpycom  23183  htpyco1  23185  htpyco2  23186  htpycc  23187  phtpycn  23190
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