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Theorem ishtpy 24488
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
ishtpy.3 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4 (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn 𝐾))
Assertion
Ref Expression
ishtpy (πœ‘ β†’ (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 Γ—t II) Cn 𝐾) ∧ βˆ€π‘  ∈ 𝑋 ((𝑠𝐻0) = (πΉβ€˜π‘ ) ∧ (𝑠𝐻1) = (πΊβ€˜π‘ )))))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   πœ‘,𝑠   𝑋,𝑠
Allowed substitution hint:   𝐾(𝑠)
Proof of Theorem ishtpy
Dummy variables 𝑓 𝑔 β„Ž 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-htpy 24486 . . . . . 6 Htpy = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (𝑓 ∈ (𝑗 Cn π‘˜), 𝑔 ∈ (𝑗 Cn π‘˜) ↦ {β„Ž ∈ ((𝑗 Γ—t II) Cn π‘˜) ∣ βˆ€π‘  ∈ βˆͺ 𝑗((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}))
21a1i 11 . . . . 5 (πœ‘ β†’ Htpy = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (𝑓 ∈ (𝑗 Cn π‘˜), 𝑔 ∈ (𝑗 Cn π‘˜) ↦ {β„Ž ∈ ((𝑗 Γ—t II) Cn π‘˜) ∣ βˆ€π‘  ∈ βˆͺ 𝑗((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))})))
3 simprl 770 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑗 = 𝐽)
4 simprr 772 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘˜ = 𝐾)
53, 4oveq12d 7427 . . . . . 6 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (𝑗 Cn π‘˜) = (𝐽 Cn 𝐾))
63oveq1d 7424 . . . . . . . 8 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (𝑗 Γ—t II) = (𝐽 Γ—t II))
76, 4oveq12d 7427 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ ((𝑗 Γ—t II) Cn π‘˜) = ((𝐽 Γ—t II) Cn 𝐾))
83unieqd 4923 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
10 toponuni 22416 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
119, 10syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
1211adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑋 = βˆͺ 𝐽)
138, 12eqtr4d 2776 . . . . . . . 8 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = 𝑋)
1413raleqdv 3326 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆ€π‘  ∈ βˆͺ 𝑗((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ )) ↔ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))))
157, 14rabeqbidv 3450 . . . . . 6 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ {β„Ž ∈ ((𝑗 Γ—t II) Cn π‘˜) ∣ βˆ€π‘  ∈ βˆͺ 𝑗((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} = {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))})
165, 5, 15mpoeq123dv 7484 . . . . 5 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (𝑓 ∈ (𝑗 Cn π‘˜), 𝑔 ∈ (𝑗 Cn π‘˜) ↦ {β„Ž ∈ ((𝑗 Γ—t II) Cn π‘˜) ∣ βˆ€π‘  ∈ βˆͺ 𝑗((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}))
17 topontop 22415 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
189, 17syl 17 . . . . 5 (πœ‘ β†’ 𝐽 ∈ Top)
19 ishtpy.3 . . . . . 6 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
20 cntop2 22745 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Top)
22 ovex 7442 . . . . . . . . . 10 ((𝐽 Γ—t II) Cn 𝐾) ∈ V
23 ssrab2 4078 . . . . . . . . . 10 {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} βŠ† ((𝐽 Γ—t II) Cn 𝐾)
2422, 23elpwi2 5347 . . . . . . . . 9 {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} ∈ 𝒫 ((𝐽 Γ—t II) Cn 𝐾)
2524rgen2w 3067 . . . . . . . 8 βˆ€π‘“ ∈ (𝐽 Cn 𝐾)βˆ€π‘” ∈ (𝐽 Cn 𝐾){β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} ∈ 𝒫 ((𝐽 Γ—t II) Cn 𝐾)
26 eqid 2733 . . . . . . . . 9 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))})
2726fmpo 8054 . . . . . . . 8 (βˆ€π‘“ ∈ (𝐽 Cn 𝐾)βˆ€π‘” ∈ (𝐽 Cn 𝐾){β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} ∈ 𝒫 ((𝐽 Γ—t II) Cn 𝐾) ↔ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}):((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾))βŸΆπ’« ((𝐽 Γ—t II) Cn 𝐾))
2825, 27mpbi 229 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}):((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾))βŸΆπ’« ((𝐽 Γ—t II) Cn 𝐾)
29 ovex 7442 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3029, 29xpex 7740 . . . . . . 7 ((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾)) ∈ V
3122pwex 5379 . . . . . . 7 𝒫 ((𝐽 Γ—t II) Cn 𝐾) ∈ V
32 fex2 7924 . . . . . . 7 (((𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}):((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾))βŸΆπ’« ((𝐽 Γ—t II) Cn 𝐾) ∧ ((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾)) ∈ V ∧ 𝒫 ((𝐽 Γ—t II) Cn 𝐾) ∈ V) β†’ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}) ∈ V)
3328, 30, 31, 32mp3an 1462 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}) ∈ V
3433a1i 11 . . . . 5 (πœ‘ β†’ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}) ∈ V)
352, 16, 18, 21, 34ovmpod 7560 . . . 4 (πœ‘ β†’ (𝐽 Htpy 𝐾) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))}))
36 fveq1 6891 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘ ) = (πΉβ€˜π‘ ))
3736eqeq2d 2744 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ↔ (π‘ β„Ž0) = (πΉβ€˜π‘ )))
38 fveq1 6891 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘”β€˜π‘ ) = (πΊβ€˜π‘ ))
3938eqeq2d 2744 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘ β„Ž1) = (π‘”β€˜π‘ ) ↔ (π‘ β„Ž1) = (πΊβ€˜π‘ )))
4037, 39bi2anan9 638 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ )) ↔ ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))))
4140adantl 483 . . . . . 6 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ )) ↔ ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))))
4241ralbidv 3178 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ )) ↔ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))))
4342rabbidv 3441 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ∧ (π‘ β„Ž1) = (π‘”β€˜π‘ ))} = {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))})
44 ishtpy.4 . . . 4 (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn 𝐾))
4522rabex 5333 . . . . 5 {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))} ∈ V
4645a1i 11 . . . 4 (πœ‘ β†’ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))} ∈ V)
4735, 43, 19, 44, 46ovmpod 7560 . . 3 (πœ‘ β†’ (𝐹(𝐽 Htpy 𝐾)𝐺) = {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))})
4847eleq2d 2820 . 2 (πœ‘ β†’ (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))}))
49 oveq 7415 . . . . . 6 (β„Ž = 𝐻 β†’ (π‘ β„Ž0) = (𝑠𝐻0))
5049eqeq1d 2735 . . . . 5 (β„Ž = 𝐻 β†’ ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ↔ (𝑠𝐻0) = (πΉβ€˜π‘ )))
51 oveq 7415 . . . . . 6 (β„Ž = 𝐻 β†’ (π‘ β„Ž1) = (𝑠𝐻1))
5251eqeq1d 2735 . . . . 5 (β„Ž = 𝐻 β†’ ((π‘ β„Ž1) = (πΊβ€˜π‘ ) ↔ (𝑠𝐻1) = (πΊβ€˜π‘ )))
5350, 52anbi12d 632 . . . 4 (β„Ž = 𝐻 β†’ (((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ )) ↔ ((𝑠𝐻0) = (πΉβ€˜π‘ ) ∧ (𝑠𝐻1) = (πΊβ€˜π‘ ))))
5453ralbidv 3178 . . 3 (β„Ž = 𝐻 β†’ (βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ )) ↔ βˆ€π‘  ∈ 𝑋 ((𝑠𝐻0) = (πΉβ€˜π‘ ) ∧ (𝑠𝐻1) = (πΊβ€˜π‘ ))))
5554elrab 3684 . 2 (𝐻 ∈ {β„Ž ∈ ((𝐽 Γ—t II) Cn 𝐾) ∣ βˆ€π‘  ∈ 𝑋 ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ∧ (π‘ β„Ž1) = (πΊβ€˜π‘ ))} ↔ (𝐻 ∈ ((𝐽 Γ—t II) Cn 𝐾) ∧ βˆ€π‘  ∈ 𝑋 ((𝑠𝐻0) = (πΉβ€˜π‘ ) ∧ (𝑠𝐻1) = (πΊβ€˜π‘ ))))
5648, 55bitrdi 287 1 (πœ‘ β†’ (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 Γ—t II) Cn 𝐾) ∧ βˆ€π‘  ∈ 𝑋 ((𝑠𝐻0) = (πΉβ€˜π‘ ) ∧ (𝑠𝐻1) = (πΊβ€˜π‘ )))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475  π’« cpw 4603  βˆͺ cuni 4909   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  0cc0 11110  1c1 11111  Topctop 22395  TopOnctopon 22412   Cn ccn 22728   Γ—t ctx 23064  IIcii 24391   Htpy chtpy 24483
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-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-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-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-htpy 24486
This theorem is referenced by:  htpycn  24489  htpyi  24490  ishtpyd  24491
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