MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ishtpy Structured version   Visualization version   GIF version

Theorem ishtpy 22579
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 22577 . . . . . 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 790 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
4 simprr 792 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 6567 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
63oveq1d 6564 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
76, 4oveq12d 6567 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
83unieqd 4382 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
10 toponuni 20542 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1211adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
138, 12eqtr4d 2647 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
1413raleqdv 3121 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))))
157, 14rabeqbidv 3168 . . . . . 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 6615 . . . . 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 20541 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
189, 17syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
19 ishtpy.3 . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
20 cntop2 20855 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
22 ssrab2 3650 . . . . . . . . . 10 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
23 ovex 6577 . . . . . . . . . . 11 ((𝐽 ×t II) Cn 𝐾) ∈ V
2423elpw2 4755 . . . . . . . . . 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 220 . . . . . . . . 9 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
2625rgen2w 2909 . . . . . . . 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 2610 . . . . . . . . 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 7126 . . . . . . . 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 219 . . . . . . 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 6577 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3130, 30xpex 6860 . . . . . . 7 ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
3223pwex 4774 . . . . . . 7 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
33 fex2 7014 . . . . . . 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 1416 . . . . . 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 6686 . . . 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 6102 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑠) = (𝐹𝑠))
3837eqeq2d 2620 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑠0) = (𝑓𝑠) ↔ (𝑠0) = (𝐹𝑠)))
39 fveq1 6102 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔𝑠) = (𝐺𝑠))
4039eqeq2d 2620 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑠1) = (𝑔𝑠) ↔ (𝑠1) = (𝐺𝑠)))
4138, 40bi2anan9 913 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4241adantl 481 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4342ralbidv 2969 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4443rabbidv 3164 . . . 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 4740 . . . . 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 6686 . . 3 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))})
4948eleq2d 2673 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))}))
50 oveq 6555 . . . . . 6 ( = 𝐻 → (𝑠0) = (𝑠𝐻0))
5150eqeq1d 2612 . . . . 5 ( = 𝐻 → ((𝑠0) = (𝐹𝑠) ↔ (𝑠𝐻0) = (𝐹𝑠)))
52 oveq 6555 . . . . . 6 ( = 𝐻 → (𝑠1) = (𝑠𝐻1))
5352eqeq1d 2612 . . . . 5 ( = 𝐻 → ((𝑠1) = (𝐺𝑠) ↔ (𝑠𝐻1) = (𝐺𝑠)))
5451, 53anbi12d 743 . . . 4 ( = 𝐻 → (((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5554ralbidv 2969 . . 3 ( = 𝐻 → (∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5655elrab 3331 . 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 275 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108   cuni 4372   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  0cc0 9815  1c1 9816  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ×t ctx 21173  IIcii 22486   Htpy chtpy 22574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-htpy 22577
This theorem is referenced by:  htpycn  22580  htpyi  22581  ishtpyd  22582
  Copyright terms: Public domain W3C validator