Theorem isphtpy 25023

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 23281	. . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3		oveq2 7400	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4		oveq2 7400	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
5	4	oveqd 7409	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
6	5	rabeqdv 3428	. . . . . . 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 7467	. . . . . 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 25013	. . . . . 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 7425	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
10	9, 9	mpoex 8056	. . . . . 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 6971	. . . . 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 18	. . . 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 7401	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14		simpl 486	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
15	14	fveq1d 6865	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
16	15	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((0ℎ𝑠) = (𝑓‘0) ↔ (0ℎ𝑠) = (𝐹‘0)))
17	14	fveq1d 6865	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
18	17	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((1ℎ𝑠) = (𝑓‘1) ↔ (1ℎ𝑠) = (𝐹‘1)))
19	16, 18	anbi12d 641	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1)) ↔ ((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))))
20	19	ralbidv 3184	. . . . . 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 3431	. . . . 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 485	. . . 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 7425	. . . . . 6 ⊢ (𝐹(II Htpy 𝐽)𝐺) ∈ V
25	24	rabex 5294	. . . . 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 7544	. . 3 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))})
28	27	eleq2d 2847	. 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 7398	. . . . . 6 ⊢ (ℎ = 𝐻 → (0ℎ𝑠) = (0𝐻𝑠))
30	29	eqeq1d 2763	. . . . 5 ⊢ (ℎ = 𝐻 → ((0ℎ𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31		oveq 7398	. . . . . 6 ⊢ (ℎ = 𝐻 → (1ℎ𝑠) = (1𝐻𝑠))
32	31	eqeq1d 2763	. . . . 5 ⊢ (ℎ = 𝐻 → ((1ℎ𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
33	30, 32	anbi12d 641	. . . 4 ⊢ (ℎ = 𝐻 → (((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
34	33	ralbidv 3184	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
35	34	elrab 3650	. 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 289	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  0cc0 11070  1c1 11071  [,]cicc 13349  Topctop 22933   Cn ccn 23264  IIcii 24917   Htpy chtpy 25009  PHtpycphtpy 25010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-map 8805  df-top 22934  df-topon 22951  df-cn 23267  df-phtpy 25013

This theorem is referenced by:  phtpyhtpy  25024  phtpyi  25026  isphtpyd  25028