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Theorem htpycn 24809
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 24808 . . 3 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ ( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)))))
5 simpl 482 . . 3 (( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))) → ∈ ((𝐽 ×t II) Cn 𝐾))
64, 5syl6bi 253 . 2 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) → ∈ ((𝐽 ×t II) Cn 𝐾)))
76ssrdv 3980 1 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  wss 3940  cfv 6533  (class class class)co 7401  0cc0 11105  1c1 11106  TopOnctopon 22722   Cn ccn 23038   ×t ctx 23374  IIcii 24705   Htpy chtpy 24803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817  df-top 22706  df-topon 22723  df-cn 23041  df-htpy 24806
This theorem is referenced by:  htpycom  24812  htpyco1  24814  htpyco2  24815  htpycc  24816  phtpycn  24819
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