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Theorem htpycn 22992
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 22991 . . 3 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ ( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)))))
5 simpl 468 . . 3 (( ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))) → ∈ ((𝐽 ×t II) Cn 𝐾))
64, 5syl6bi 243 . 2 (𝜑 → ( ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) → ∈ ((𝐽 ×t II) Cn 𝐾)))
76ssrdv 3758 1 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  cfv 6030  (class class class)co 6796  0cc0 10142  1c1 10143  TopOnctopon 20935   Cn ccn 21249   ×t ctx 21584  IIcii 22898   Htpy chtpy 22986
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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015  df-top 20919  df-topon 20936  df-cn 21252  df-htpy 22989
This theorem is referenced by:  htpycom  22995  htpyco1  22997  htpyco2  22998  htpycc  22999  phtpycn  23002
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