Theorem htpycn 24940

Description: A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)

Hypotheses

Ref	Expression
ishtpy.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ishtpy.3	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4	⊢ (𝜑 → 𝐺 ∈ (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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishtpy.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		ishtpy.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		ishtpy.4	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
4	1, 2, 3	ishtpy 24939	. . 3 ⊢ (𝜑 → (ℎ ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)))))
5		simpl 482	. . 3 ⊢ ((ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))) → ℎ ∈ ((𝐽 ×t II) Cn 𝐾))
6	4, 5	biimtrdi 253	. 2 ⊢ (𝜑 → (ℎ ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) → ℎ ∈ ((𝐽 ×t II) Cn 𝐾)))
7	6	ssrdv 3927	1 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039  TopOnctopon 22875   Cn ccn 23189   ×_t ctx 23525  IIcii 24842   Htpy chtpy 24934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-top 22859  df-topon 22876  df-cn 23192  df-htpy 24937

This theorem is referenced by:  htpycom  24943  htpyco1  24945  htpyco2  24946  htpycc  24947  phtpycn  24950