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Theorem ishtpy 23580
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 23578 . . . . . 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 770 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
4 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 7157 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
63oveq1d 7154 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
76, 4oveq12d 7157 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
83unieqd 4817 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
10 toponuni 21522 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1211adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
138, 12eqtr4d 2839 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
1413raleqdv 3367 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑠 𝑗((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))))
157, 14rabeqbidv 3436 . . . . . 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 7212 . . . . 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 21521 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
189, 17syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
19 ishtpy.3 . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
20 cntop2 21849 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
22 ovex 7172 . . . . . . . . . 10 ((𝐽 ×t II) Cn 𝐾) ∈ V
23 ssrab2 4010 . . . . . . . . . 10 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
2422, 23elpwi2 5216 . . . . . . . . 9 { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
2524rgen2w 3122 . . . . . . . 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 2801 . . . . . . . . 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 7752 . . . . . . . 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 233 . . . . . . 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 7172 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3029, 29xpex 7460 . . . . . . 7 ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
3122pwex 5249 . . . . . . 7 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32 fex2 7624 . . . . . . 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 1458 . . . . . 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 7285 . . . 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 6648 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑠) = (𝐹𝑠))
3736eqeq2d 2812 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑠0) = (𝑓𝑠) ↔ (𝑠0) = (𝐹𝑠)))
38 fveq1 6648 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔𝑠) = (𝐺𝑠))
3938eqeq2d 2812 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑠1) = (𝑔𝑠) ↔ (𝑠1) = (𝐺𝑠)))
4037, 39bi2anan9 638 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4140adantl 485 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4241ralbidv 3165 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (∀𝑠𝑋 ((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠)) ↔ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))))
4342rabbidv 3430 . . . 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 5202 . . . . 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 7285 . . 3 (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))})
4847eleq2d 2878 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))}))
49 oveq 7145 . . . . . 6 ( = 𝐻 → (𝑠0) = (𝑠𝐻0))
5049eqeq1d 2803 . . . . 5 ( = 𝐻 → ((𝑠0) = (𝐹𝑠) ↔ (𝑠𝐻0) = (𝐹𝑠)))
51 oveq 7145 . . . . . 6 ( = 𝐻 → (𝑠1) = (𝑠𝐻1))
5251eqeq1d 2803 . . . . 5 ( = 𝐻 → ((𝑠1) = (𝐺𝑠) ↔ (𝑠𝐻1) = (𝐺𝑠)))
5350, 52anbi12d 633 . . . 4 ( = 𝐻 → (((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5453ralbidv 3165 . . 3 ( = 𝐻 → (∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠)) ↔ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5554elrab 3631 . 2 (𝐻 ∈ { ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠𝑋 ((𝑠0) = (𝐹𝑠) ∧ (𝑠1) = (𝐺𝑠))} ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
5648, 55syl6bb 290 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 209  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  𝒫 cpw 4500   cuni 4803   × cxp 5521  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  0cc0 10530  1c1 10531  Topctop 21501  TopOnctopon 21518   Cn ccn 21832   ×t ctx 22168  IIcii 23483   Htpy chtpy 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-top 21502  df-topon 21519  df-cn 21835  df-htpy 23578
This theorem is referenced by:  htpycn  23581  htpyi  23582  ishtpyd  23583
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