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Theorem phtpyhtpy 25028
Description: A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2 (𝜑𝐹 ∈ (II Cn 𝐽))
isphtpy.3 (𝜑𝐺 ∈ (II Cn 𝐽))
Assertion
Ref Expression
phtpyhtpy (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Proof of Theorem phtpyhtpy
Dummy variables 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isphtpy.2 . . . 4 (𝜑𝐹 ∈ (II Cn 𝐽))
2 isphtpy.3 . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
31, 2isphtpy 25027 . . 3 (𝜑 → ( ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ ( ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)))))
4 simpl 482 . . 3 (( ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))) → ∈ (𝐹(II Htpy 𝐽)𝐺))
53, 4biimtrdi 253 . 2 (𝜑 → ( ∈ (𝐹(PHtpy‘𝐽)𝐺) → ∈ (𝐹(II Htpy 𝐽)𝐺)))
65ssrdv 4001 1 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  [,]cicc 13387   Cn ccn 23248  IIcii 24915   Htpy chtpy 25013  PHtpycphtpy 25014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-phtpy 25017
This theorem is referenced by:  phtpycn  25029  phtpy01  25031  phtpycom  25034  phtpyco2  25036  phtpycc  25037  pcohtpylem  25066  txsconnlem  35225  cvmliftphtlem  35302
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