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Definition df-phtpc 22710
Description: Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
Assertion
Ref Expression
df-phtpc ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
Distinct variable group:   𝑥,𝑓,𝑔

Detailed syntax breakdown of Definition df-phtpc
StepHypRef Expression
1 cphtpc 22687 . 2 class ph
2 vx . . 3 setvar 𝑥
3 ctop 20626 . . 3 class Top
4 vf . . . . . . . 8 setvar 𝑓
54cv 1479 . . . . . . 7 class 𝑓
6 vg . . . . . . . 8 setvar 𝑔
76cv 1479 . . . . . . 7 class 𝑔
85, 7cpr 4155 . . . . . 6 class {𝑓, 𝑔}
9 cii 22597 . . . . . . 7 class II
102cv 1479 . . . . . . 7 class 𝑥
11 ccn 20947 . . . . . . 7 class Cn
129, 10, 11co 6610 . . . . . 6 class (II Cn 𝑥)
138, 12wss 3559 . . . . 5 wff {𝑓, 𝑔} ⊆ (II Cn 𝑥)
14 cphtpy 22686 . . . . . . . 8 class PHtpy
1510, 14cfv 5852 . . . . . . 7 class (PHtpy‘𝑥)
165, 7, 15co 6610 . . . . . 6 class (𝑓(PHtpy‘𝑥)𝑔)
17 c0 3896 . . . . . 6 class
1816, 17wne 2790 . . . . 5 wff (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅
1913, 18wa 384 . . . 4 wff ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)
2019, 4, 6copab 4677 . . 3 class {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)}
212, 3, 20cmpt 4678 . 2 class (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
221, 21wceq 1480 1 wff ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
Colors of variables: wff setvar class
This definition is referenced by:  phtpcrel  22711  isphtpc  22712
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