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Theorem ishtpyd 22994
Description: Deduction for membership in the class of homotopies. (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))
ishtpyd.1 (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
ishtpyd.2 ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))
ishtpyd.3 ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))
Assertion
Ref Expression
ishtpyd (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠   𝑋,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem ishtpyd
StepHypRef Expression
1 ishtpyd.1 . 2 (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
2 ishtpyd.2 . . . 4 ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))
3 ishtpyd.3 . . . 4 ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))
42, 3jca 501 . . 3 ((𝜑𝑠𝑋) → ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
54ralrimiva 3115 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
6 ishtpy.1 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
7 ishtpy.3 . . 3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
8 ishtpy.4 . . 3 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
96, 7, 8ishtpy 22991 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
101, 5, 9mpbir2and 692 1 (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  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  htpyid  22996  htpyco1  22997  htpyco2  22998  htpycc  22999  isphtpy2d  23006
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