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Theorem ishtpy 24287
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 24285 . . . . . 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 769 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑗 = 𝐽)
4 simprr 771 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘˜ = 𝐾)
53, 4oveq12d 7369 . . . . . 6 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (𝑗 Cn π‘˜) = (𝐽 Cn 𝐾))
63oveq1d 7366 . . . . . . . 8 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (𝑗 Γ—t II) = (𝐽 Γ—t II))
76, 4oveq12d 7369 . . . . . . 7 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ ((𝑗 Γ—t II) Cn π‘˜) = ((𝐽 Γ—t II) Cn 𝐾))
83unieqd 4877 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
9 ishtpy.1 . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
10 toponuni 22215 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
119, 10syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
1211adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑋 = βˆͺ 𝐽)
138, 12eqtr4d 2780 . . . . . . . 8 ((πœ‘ ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = 𝑋)
1413raleqdv 3311 . . . . . . 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 3422 . . . . . 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 7426 . . . . 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 22214 . . . . . 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 22544 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
2119, 20syl 17 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Top)
22 ovex 7384 . . . . . . . . . 10 ((𝐽 Γ—t II) Cn 𝐾) ∈ V
23 ssrab2 4035 . . . . . . . . . 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 5301 . . . . . . . . 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 2737 . . . . . . . . 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 7992 . . . . . . . 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 7384 . . . . . . . 8 (𝐽 Cn 𝐾) ∈ V
3029, 29xpex 7679 . . . . . . 7 ((𝐽 Cn 𝐾) Γ— (𝐽 Cn 𝐾)) ∈ V
3122pwex 5333 . . . . . . 7 𝒫 ((𝐽 Γ—t II) Cn 𝐾) ∈ V
32 fex2 7862 . . . . . . 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 1461 . . . . . 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 7501 . . . 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 6838 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘ ) = (πΉβ€˜π‘ ))
3736eqeq2d 2748 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘ β„Ž0) = (π‘“β€˜π‘ ) ↔ (π‘ β„Ž0) = (πΉβ€˜π‘ )))
38 fveq1 6838 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘”β€˜π‘ ) = (πΊβ€˜π‘ ))
3938eqeq2d 2748 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘ β„Ž1) = (π‘”β€˜π‘ ) ↔ (π‘ β„Ž1) = (πΊβ€˜π‘ )))
4037, 39bi2anan9 637 . . . . . . 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 482 . . . . . 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 3172 . . . . 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 3413 . . . 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 5287 . . . . 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 7501 . . 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 2823 . 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 7357 . . . . . 6 (β„Ž = 𝐻 β†’ (π‘ β„Ž0) = (𝑠𝐻0))
5049eqeq1d 2739 . . . . 5 (β„Ž = 𝐻 β†’ ((π‘ β„Ž0) = (πΉβ€˜π‘ ) ↔ (𝑠𝐻0) = (πΉβ€˜π‘ )))
51 oveq 7357 . . . . . 6 (β„Ž = 𝐻 β†’ (π‘ β„Ž1) = (𝑠𝐻1))
5251eqeq1d 2739 . . . . 5 (β„Ž = 𝐻 β†’ ((π‘ β„Ž1) = (πΊβ€˜π‘ ) ↔ (𝑠𝐻1) = (πΊβ€˜π‘ )))
5350, 52anbi12d 631 . . . 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 3172 . . 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 3643 . 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 286 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3062  {crab 3405  Vcvv 3443  π’« cpw 4558  βˆͺ cuni 4863   Γ— cxp 5629  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  0cc0 11009  1c1 11010  Topctop 22194  TopOnctopon 22211   Cn ccn 22527   Γ—t ctx 22863  IIcii 24190   Htpy chtpy 24282
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-map 8725  df-top 22195  df-topon 22212  df-cn 22530  df-htpy 24285
This theorem is referenced by:  htpycn  24288  htpyi  24289  ishtpyd  24290
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