Theorem isphtpy 24497

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 22745	. . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3		oveq2 7417	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4		oveq2 7417	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
5	4	oveqd 7426	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
6	5	rabeqdv 3448	. . . . . . 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 7484	. . . . . 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 24487	. . . . . 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 7442	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
10	9, 9	mpoex 8066	. . . . . 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 6999	. . . . 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 7418	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14		simpl 484	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
15	14	fveq1d 6894	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
16	15	eqeq2d 2744	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((0ℎ𝑠) = (𝑓‘0) ↔ (0ℎ𝑠) = (𝐹‘0)))
17	14	fveq1d 6894	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
18	17	eqeq2d 2744	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((1ℎ𝑠) = (𝑓‘1) ↔ (1ℎ𝑠) = (𝐹‘1)))
19	16, 18	anbi12d 632	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1)) ↔ ((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))))
20	19	ralbidv 3178	. . . . . 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 3450	. . . . 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 483	. . . 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 7442	. . . . . 6 ⊢ (𝐹(II Htpy 𝐽)𝐺) ∈ V
25	24	rabex 5333	. . . . 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 7560	. . 3 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = {ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))})
28	27	eleq2d 2820	. 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 7415	. . . . . 6 ⊢ (ℎ = 𝐻 → (0ℎ𝑠) = (0𝐻𝑠))
30	29	eqeq1d 2735	. . . . 5 ⊢ (ℎ = 𝐻 → ((0ℎ𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31		oveq 7415	. . . . . 6 ⊢ (ℎ = 𝐻 → (1ℎ𝑠) = (1𝐻𝑠))
32	31	eqeq1d 2735	. . . . 5 ⊢ (ℎ = 𝐻 → ((1ℎ𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
33	30, 32	anbi12d 632	. . . 4 ⊢ (ℎ = 𝐻 → (((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
34	33	ralbidv 3178	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
35	34	elrab 3684	. 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 287	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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  0cc0 11110  1c1 11111  [,]cicc 13327  Topctop 22395   Cn ccn 22728  IIcii 24391   Htpy chtpy 24483  PHtpycphtpy 24484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-phtpy 24487

This theorem is referenced by:  phtpyhtpy  24498  phtpyi  24500  isphtpyd  24502