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Theorem ishtpy 24145
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 24143 . . . . . 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 768 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
4 simprr 770 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 7285 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
63oveq1d 7282 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
76, 4oveq12d 7285 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
83unieqd 4853 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
10 toponuni 22073 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1211adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
138, 12eqtr4d 2781 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
1413raleqdv 3346 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))))
157, 14rabeqbidv 3417 . . . . . 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, 15mpoeq123dv 7340 . . . . 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 22072 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
189, 17syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
19 ishtpy.3 . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
20 cntop2 22402 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
22 ovex 7300 . . . . . . . . . 10 ((𝐽 ×t II) Cn 𝐾) ∈ V
23 ssrab2 4012 . . . . . . . . . 10 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
2422, 23elpwi2 5268 . . . . . . . . 9 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
2524rgen2w 3077 . . . . . . . 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)
26 eqid 2738 . . . . . . . . 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))})
2726fmpo 7897 . . . . . . . 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))
2825, 27mpbi 229 . . . . . . 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)
29 ovex 7300 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3029, 29xpex 7593 . . . . . . 7 ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
3122pwex 5301 . . . . . . 7 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32 fex2 7770 . . . . . . 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)
3328, 30, 31, 32mp3an 1460 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) ∈ V
3433a1i 11 . . . . 5 (𝜑 → (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}) ∈ V)
352, 16, 18, 21, 34ovmpod 7415 . . . 4 (𝜑 → (𝐽 Htpy 𝐾) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
36 fveq1 6765 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑠) = (𝐹𝑠))
3736eqeq2d 2749 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑠0) = (𝑓𝑠) ↔ (𝑠0) = (𝐹𝑠)))
38 fveq1 6765 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔𝑠) = (𝐺𝑠))
3938eqeq2d 2749 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑠1) = (𝑔𝑠) ↔ (𝑠1) = (𝐺𝑠)))
4037, 39bi2anan9 636 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4140adantl 482 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4241ralbidv 3121 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4342rabbidv 3411 . . . 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))})
44 ishtpy.4 . . . 4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
4522rabex 5254 . . . . 5 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ∈ V
4645a1i 11 . . . 4 (𝜑 → { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ∈ V)
4735, 43, 19, 44, 46ovmpod 7415 . . 3 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))})
4847eleq2d 2824 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))}))
49 oveq 7273 . . . . . 6 ( = 𝐻 → (𝑠0) = (𝑠𝐻0))
5049eqeq1d 2740 . . . . 5 ( = 𝐻 → ((𝑠0) = (𝐹𝑠) ↔ (𝑠𝐻0) = (𝐹𝑠)))
51 oveq 7273 . . . . . 6 ( = 𝐻 → (𝑠1) = (𝑠𝐻1))
5251eqeq1d 2740 . . . . 5 ( = 𝐻 → ((𝑠1) = (𝐺𝑠) ↔ (𝑠𝐻1) = (𝐺𝑠)))
5350, 52anbi12d 631 . . . 4 ( = 𝐻 → (((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5453ralbidv 3121 . . 3 ( = 𝐻 → (∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5554elrab 3623 . 2 (𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5648, 55bitrdi 287 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3429  𝒫 cpw 4533   cuni 4839   × cxp 5582  wf 6422  cfv 6426  (class class class)co 7267  cmpo 7269  0cc0 10881  1c1 10882  Topctop 22052  TopOnctopon 22069   Cn ccn 22385   ×t ctx 22721  IIcii 24048   Htpy chtpy 24140
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-1st 7820  df-2nd 7821  df-map 8604  df-top 22053  df-topon 22070  df-cn 22388  df-htpy 24143
This theorem is referenced by:  htpycn  24146  htpyi  24147  ishtpyd  24148
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