Theorem isphtpy 25101

Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
isphtpy.2	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
isphtpy.3	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))

Assertion

Ref	Expression
isphtpy	⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))

Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠

Proof of Theorem isphtpy

Dummy variables 𝑓 𝑔 ℎ 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isphtpy.2	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
2		cntop2 23359	. . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3		oveq2 7408	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4		oveq2 7408	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
5	4	oveqd 7417	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
6	5	rabeqdv 3432	. . . . . . 7 ⊢ (𝑗 = 𝐽 → {ℎ ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))} = {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))})
7	3, 3, 6	mpoeq123dv 7475	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ {ℎ ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
8		df-phtpy 25091	. . . . . 6 ⊢ PHtpy = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ {ℎ ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
9		ovex 7433	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
10	9, 9	mpoex 8064	. . . . . 6 ⊢ (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}) ∈ V
11	7, 8, 10	fvmpt 6979	. . . . 5 ⊢ (𝐽 ∈ Top → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
12	1, 2, 11	3syl 19	. . . 4 ⊢ (𝜑 → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
13		oveq12 7409	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14		simpl 487	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
15	14	fveq1d 6873	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
16	15	eqeq2d 2776	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((0ℎ𝑠) = (𝑓‘0) ↔ (0ℎ𝑠) = (𝐹‘0)))
17	14	fveq1d 6873	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
18	17	eqeq2d 2776	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((1ℎ𝑠) = (𝑓‘1) ↔ (1ℎ𝑠) = (𝐹‘1)))
19	16, 18	anbi12d 643	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1)) ↔ ((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))))
20	19	ralbidv 3188	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))))
21	13, 20	rabeqbidv 3435	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))} = {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))})
22	21	adantl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → {ℎ ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))} = {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))})
23		isphtpy.3	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
24		ovex 7433	. . . . . 6 ⊢ (𝐹(II Htpy 𝐽)𝐺) ∈ V
25	24	rabex 5300	. . . . 5 ⊢ {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))} ∈ V
26	25	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))} ∈ V)
27	12, 22, 1, 23, 26	ovmpod 7552	. . 3 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))})
28	27	eleq2d 2851	. 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))}))
29		oveq 7406	. . . . . 6 ⊢ (ℎ = 𝐻 → (0ℎ𝑠) = (0𝐻𝑠))
30	29	eqeq1d 2767	. . . . 5 ⊢ (ℎ = 𝐻 → ((0ℎ𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31		oveq 7406	. . . . . 6 ⊢ (ℎ = 𝐻 → (1ℎ𝑠) = (1𝐻𝑠))
32	31	eqeq1d 2767	. . . . 5 ⊢ (ℎ = 𝐻 → ((1ℎ𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
33	30, 32	anbi12d 643	. . . 4 ⊢ (ℎ = 𝐻 → (((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
34	33	ralbidv 3188	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
35	34	elrab 3653	. 2 ⊢ (𝐻 ∈ {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))} ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
36	28, 35	bitrdi 290	1 ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417  Vcvv 3457  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  0cc0 11088  1c1 11089  [,]cicc 13366  Topctop 23011   Cn ccn 23342  IIcii 24995   Htpy chtpy 25087  PHtpycphtpy 25088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-top 23012  df-topon 23029  df-cn 23345  df-phtpy 25091

This theorem is referenced by:  phtpyhtpy  25102  phtpyi  25104  isphtpyd  25106