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Theorem htpycn 23012
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 23011 . . 3 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ ( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)))))
5 simpl 469 . . 3 (( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))) → ∈ ((𝐽 ×t II) Cn 𝐾))
64, 5syl6bi 244 . 2 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) → ∈ ((𝐽 ×t II) Cn 𝐾)))
76ssrdv 3764 1 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1634  wcel 2148  wral 3064  wss 3729  cfv 6042  (class class class)co 6812  0cc0 10159  1c1 10160  TopOnctopon 20955   Cn ccn 21269   ×t ctx 21604  IIcii 22918   Htpy chtpy 23006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-1st 7336  df-2nd 7337  df-map 8032  df-top 20939  df-topon 20956  df-cn 21272  df-htpy 23009
This theorem is referenced by:  htpycom  23015  htpyco1  23017  htpyco2  23018  htpycc  23019  phtpycn  23022
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