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Theorem htpycn 24923

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 24922	. . 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 3964	1 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ⊆ wss 3926 ‘cfv 6531 (class class class)co 7405 0cc0 11129 1c1 11130 TopOnctopon 22848 Cn ccn 23162 ×_t ctx 23498 IIcii 24819 Htpy chtpy 24917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-map 8842 df-top 22832 df-topon 22849 df-cn 23165 df-htpy 24920

This theorem is referenced by: htpycom 24926 htpyco1 24928 htpyco2 24929 htpycc 24930 phtpycn 24933

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