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Theorem ishtpy 22964
Description: Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
Hypotheses
Ref Expression
ishtpy.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
ishtpy.3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
Assertion
Ref Expression
ishtpy (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠   𝑋,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem ishtpy
Dummy variables 𝑓 𝑔 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-htpy 22962 . . . . . 6 Htpy = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ { ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
21a1i 11 . . . . 5 (𝜑 → Htpy = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ { ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))})))
3 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
4 simprr 813 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 6823 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
63oveq1d 6820 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
76, 4oveq12d 6823 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
83unieqd 4590 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
10 toponuni 20913 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1211adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
138, 12eqtr4d 2789 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
1413raleqdv 3275 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))))
157, 14rabeqbidv 3327 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → { ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))})
165, 5, 15mpt2eq123dv 6874 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ { ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
17 topontop 20912 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
189, 17syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
19 ishtpy.3 . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
20 cntop2 21239 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
22 ssrab2 3820 . . . . . . . . . 10 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
23 ovex 6833 . . . . . . . . . . 11 ((𝐽 ×t II) Cn 𝐾) ∈ V
2423elpw2 4969 . . . . . . . . . 10 ({ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾) ↔ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾))
2522, 24mpbir 221 . . . . . . . . 9 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
2625rgen2w 3055 . . . . . . . 8 𝑓 ∈ (𝐽 Cn 𝐾)∀𝑔 ∈ (𝐽 Cn 𝐾){ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
27 eqid 2752 . . . . . . . . 9 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))})
2827fmpt2 7397 . . . . . . . 8 (∀𝑓 ∈ (𝐽 Cn 𝐾)∀𝑔 ∈ (𝐽 Cn 𝐾){ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾) ↔ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾))
2926, 28mpbi 220 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾)
30 ovex 6833 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3130, 30xpex 7119 . . . . . . 7 ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
3223pwex 4989 . . . . . . 7 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
33 fex2 7278 . . . . . . 7 (((𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾) ∧ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V ∧ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V) → (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) ∈ V)
3429, 31, 32, 33mp3an 1565 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) ∈ V
3534a1i 11 . . . . 5 (𝜑 → (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) ∈ V)
362, 16, 18, 21, 35ovmpt2d 6945 . . . 4 (𝜑 → (𝐽 Htpy 𝐾) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
37 fveq1 6343 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑠) = (𝐹𝑠))
3837eqeq2d 2762 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑠0) = (𝑓𝑠) ↔ (𝑠0) = (𝐹𝑠)))
39 fveq1 6343 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔𝑠) = (𝐺𝑠))
4039eqeq2d 2762 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑠1) = (𝑔𝑠) ↔ (𝑠1) = (𝐺𝑠)))
4138, 40bi2anan9 953 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4241adantl 473 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4342ralbidv 3116 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4443rabbidv 3321 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))})
45 ishtpy.4 . . . 4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
4623rabex 4956 . . . . 5 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ∈ V
4746a1i 11 . . . 4 (𝜑 → { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ∈ V)
4836, 44, 19, 45, 47ovmpt2d 6945 . . 3 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))})
4948eleq2d 2817 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))}))
50 oveq 6811 . . . . . 6 ( = 𝐻 → (𝑠0) = (𝑠𝐻0))
5150eqeq1d 2754 . . . . 5 ( = 𝐻 → ((𝑠0) = (𝐹𝑠) ↔ (𝑠𝐻0) = (𝐹𝑠)))
52 oveq 6811 . . . . . 6 ( = 𝐻 → (𝑠1) = (𝑠𝐻1))
5352eqeq1d 2754 . . . . 5 ( = 𝐻 → ((𝑠1) = (𝐺𝑠) ↔ (𝑠𝐻1) = (𝐺𝑠)))
5451, 53anbi12d 749 . . . 4 ( = 𝐻 → (((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5554ralbidv 3116 . . 3 ( = 𝐻 → (∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5655elrab 3496 . 2 (𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5749, 56syl6bb 276 1 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  {crab 3046  Vcvv 3332  wss 3707  𝒫 cpw 4294   cuni 4580   × cxp 5256  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  0cc0 10120  1c1 10121  Topctop 20892  TopOnctopon 20909   Cn ccn 21222   ×t ctx 21557  IIcii 22871   Htpy chtpy 22959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-map 8017  df-top 20893  df-topon 20910  df-cn 21225  df-htpy 22962
This theorem is referenced by:  htpycn  22965  htpyi  22966  ishtpyd  22967
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