P. 357 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cvmliftlem3 35601*	Lemma for cvmlift 35613. Since 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ 𝑊), every element 𝐴 ∈ 𝑊 satisfies (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀)). (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))    &   ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀)))
Theorem	cvmliftlem4 35602*	Lemma for cvmlift 35613. The function 𝑄 will be our lifted path, defined piecewise on each section [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] for 𝑀 ∈ (1...𝑁). For 𝑀 = 0, it is a "seed" value which makes the rest of the recursion work, a singleton function mapping 0 to 𝑃. (Contributed by Mario Carneiro, 15-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    ⇒   ⊢ (𝑄‘0) = {⟨0, 𝑃⟩}
Theorem	cvmliftlem5 35603*	Lemma for cvmlift 35613. Definition of 𝑄 at a successor. This is a function defined on 𝑊 as ◡(𝑇 ↾ 𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2nd ‘(𝑇‘𝑀) that contains 𝑄(𝑀 − 1) evaluated at the last defined point, namely (𝑀 − 1) / 𝑁 (note that for 𝑀 = 1 this is using the seed value 𝑄(0)(0) = 𝑃). (Contributed by Mario Carneiro, 15-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
Theorem	cvmliftlem6 35604*	Lemma for cvmlift 35613. Induction step for cvmliftlem7 35605. Assuming that 𝑄(𝑀 − 1) is defined at (𝑀 − 1) / 𝑁 and is a preimage of 𝐺((𝑀 − 1) / 𝑁), the next segment 𝑄(𝑀) is also defined and is a function on 𝑊 which is a lift 𝐺 for this segment. This follows explicitly from the definition 𝑄(𝑀) = ◡(𝐹 ↾ 𝐼) ∘ 𝐺 since 𝐺 is in 1st ‘(𝐹‘𝑀) for the entire interval so that ◡(𝐹 ↾ 𝐼) maps this into 𝐼 and 𝐹 ∘ 𝑄 maps back to 𝐺. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊)))
Theorem	cvmliftlem7 35605*	Lemma for cvmlift 35613. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 35604 to show functionality and lifting of 𝑄). (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
Theorem	cvmliftlem8 35606*	Lemma for cvmlift 35613. The functions 𝑄 are continuous functions because they are defined as ◡(𝐹 ↾ 𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹 ↾ 𝐼) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))
Theorem	cvmliftlem9 35607*	Lemma for cvmlift 35613. The 𝑄(𝑀) functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the 𝑄 functions agree on their common domain. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘𝑀)‘((𝑀 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
Theorem	cvmliftlem10 35608*	Lemma for cvmlift 35613. The function 𝐾 is going to be our complete lifted path, formed by unioning together all the 𝑄 functions (each of which is defined on one segment [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] of the interval). Here we prove by induction that 𝐾 is a continuous function and a lift of 𝐺 by applying cvmliftlem6 35604, cvmliftlem7 35605 (to show it is a function and a lift), cvmliftlem8 35606 (to show it is continuous), and cvmliftlem9 35607 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 23334 gives that 𝐾 is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)    &   ⊢ (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))    ⇒   ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
Theorem	cvmliftlem11 35609*	Lemma for cvmlift 35613. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)    ⇒   ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺))
Theorem	cvmliftlem13 35610*	Lemma for cvmlift 35613. The initial value of 𝐾 is 𝑃 because 𝑄(1) is a subset of 𝐾 which takes value 𝑃 at 0. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)    ⇒   ⊢ (𝜑 → (𝐾‘0) = 𝑃)
Theorem	cvmliftlem14 35611*	Lemma for cvmlift 35613. Putting the results of cvmliftlem11 35609, cvmliftlem13 35610 and cvmliftmo 35598 together, we have that 𝐾 is a continuous function, satisfies 𝐹 ∘ 𝐾 = 𝐺 and 𝐾(0) = 𝑃, and is equal to any other function which also has these properties, so it follows that 𝐾 is the unique lift of 𝐺. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmliftlem15 35612*	Lemma for cvmlift 35613. Discharge the assumptions of cvmliftlem14 35611. The set of all open subsets 𝑢 of the unit interval such that 𝐺 “ 𝑢 is contained in an even covering of some open set in 𝐽 is a cover of II by the definition of a covering map, so by the Lebesgue number lemma lebnumii 25008, there is a subdivision of the closed unit interval into 𝑁 equal parts such that each part is entirely contained within one such open set of 𝐽. Then using finite choice ac6sfi 9224 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35611. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmlift 35613*	One of the important properties of covering maps is that any path 𝐺 in the base space "lifts" to a path 𝑓 in the covering space such that 𝐹 ∘ 𝑓 = 𝐺, and given a starting point 𝑃 in the covering space this lift is unique. The proof is contained in cvmliftlem1 35599 thru cvmliftlem15 35612. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    ⇒   ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmfo 35614	A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)
Theorem	cvmliftiota 35615*	Write out a function 𝐻 that is the unique lift of 𝐹. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))
Theorem	cvmlift2lem1 35616*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ (∀𝑦 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑦})((𝑢 × {𝑥}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑥}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀))
Theorem	cvmlift2lem9a 35617*	Lemma for cvmlift2 35630 and cvmlift3 35642. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))    &   ⊢ (𝜑 → 𝑀 ⊆ 𝑌)    &   ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)    ⇒   ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
Theorem	cvmlift2lem2 35618*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
Theorem	cvmlift2lem3 35619*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋)))
Theorem	cvmlift2lem4 35620*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌))
Theorem	cvmlift2lem5 35621*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
Theorem	cvmlift2lem6 35622*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×t II) ↾t ({𝑋} × (0[,]1))) Cn 𝐶))
Theorem	cvmlift2lem7 35623*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺)
Theorem	cvmlift2lem8 35624*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑋𝐾0) = (𝐻‘𝑋))
Theorem	cvmlift2lem9 35625*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝑀))    &   ⊢ (𝜑 → 𝑈 ∈ II)    &   ⊢ (𝜑 → 𝑉 ∈ II)    &   ⊢ (𝜑 → (II ↾t 𝑈) ∈ Conn)    &   ⊢ (𝜑 → (II ↾t 𝑉) ∈ Conn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (◡𝐺 “ 𝑀))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×t II) ↾t (𝑈 × {𝑍})) Cn 𝐶))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝑋𝐾𝑌) ∈ 𝑏)    ⇒   ⊢ (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶))
Theorem	cvmlift2lem10 35626*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → 𝑋 ∈ (0[,]1))    &   ⊢ (𝜑 → 𝑌 ∈ (0[,]1))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
Theorem	cvmlift2lem11 35627*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}    &   ⊢ (𝜑 → 𝑈 ∈ II)    &   ⊢ (𝜑 → 𝑉 ∈ II)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×t II) ↾t (𝑈 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶)))    ⇒   ⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀))
Theorem	cvmlift2lem12 35628*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}    &   ⊢ 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀}    &   ⊢ 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))}    ⇒   ⊢ (𝜑 → 𝐾 ∈ ((II ×t II) Cn 𝐶))
Theorem	cvmlift2lem13 35629*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))
Theorem	cvmlift2 35630*	A two-dimensional version of cvmlift 35613. There is a unique lift of functions on the unit square II ×t II which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (0𝑓0) = 𝑃))
Theorem	cvmliftphtlem 35631*	Lemma for cvmliftpht 35632. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))    &   ⊢ (𝜑 → 𝐴 ∈ ((II ×t II) Cn 𝐶))    &   ⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)    &   ⊢ (𝜑 → (0𝐴0) = 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Theorem	cvmliftpht 35632*	If 𝐺 and 𝐻 are path-homotopic, then their lifts 𝑀 and 𝑁 are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐻)    ⇒   ⊢ (𝜑 → 𝑀( ≃ph‘𝐶)𝑁)
Theorem	cvmlift3lem1 35633*	Lemma for cvmlift3 35642. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑀‘0) = 𝑂)    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑁‘0) = 𝑂)    &   ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))    ⇒   ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))
Theorem	cvmlift3lem2 35634*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃!𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
Theorem	cvmlift3lem3 35635*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
Theorem	cvmlift3lem4 35636*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
Theorem	cvmlift3lem5 35637*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
Theorem	cvmlift3lem6 35638*	Lemma for cvmlift3 35642. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ (II Cn 𝐾))    &   ⊢ 𝑅 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑄) ∧ (𝑔‘0) = 𝑃))    &   ⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn (𝐾 ↾t 𝑀)))    &   ⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))    &   ⊢ 𝐼 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = (𝐻‘𝑋)))    ⇒   ⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)
Theorem	cvmlift3lem7 35639*	Lemma for cvmlift3 35642. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ PConn)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Theorem	cvmlift3lem8 35640*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝐾 Cn 𝐶))
Theorem	cvmlift3lem9 35641*	Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
Theorem	cvmlift3 35642*	A general version of cvmlift 35613. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
21.6.9  Normal numbers
Theorem	snmlff 35643*	The function 𝐹 from snmlval 35645 is a mapping from positive integers to real numbers in the range [0, 1]. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ 𝐹:ℕ⟶(0[,]1)
Theorem	snmlfval 35644*	The function 𝐹 from snmlval 35645 maps 𝑁 to the relative density of 𝐵 in the first 𝑁 digits of the digit string of 𝐴 in base 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
Theorem	snmlval 35645*	The property "𝐴 is simply normal in base 𝑅". A number is simply normal if each digit 0 ≤ 𝑏 < 𝑅 occurs in the base- 𝑅 digit string of 𝐴 with frequency 1 / 𝑅 (which is consistent with the expectation in an infinite random string of numbers selected from 0...𝑅 − 1). (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑆 = (𝑟 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    ⇒   ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
Theorem	snmlflim 35646*	If 𝐴 is simply normal, then the function 𝐹 of relative density of 𝐵 in the digit string converges to 1 / 𝑅, i.e. the set of occurrences of 𝐵 in the digit string has natural density 1 / 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑆 = (𝑟 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ ((𝐴 ∈ (𝑆‘𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))
21.6.10  Godel-sets of formulas - part 1
Syntax	cgoe 35647	The Godel-set of membership.
class ∈𝑔
Syntax	cgna 35648	The Godel-set for the Sheffer stroke.
class ⊼𝑔
Syntax	cgol 35649	The Godel-set of universal quantification. (Note that this is not a wff.)
class ∀𝑔𝑁𝑈
Syntax	csat 35650	The satisfaction function.
class Sat
Syntax	cfmla 35651	The formula set predicate.
class Fmla
Syntax	csate 35652	The ∈-satisfaction function.
class Sat∈
Syntax	cprv 35653	The "proves" relation.
class ⊧
Definition	df-goel 35654	Define the Godel-set of membership. Here the arguments 𝑥 = ⟨𝑁, 𝑃⟩ correspond to vN and vP , so (∅∈𝑔1o) actually means v0 ∈ v1 , not 0 ∈ 1. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
Definition	df-gona 35655	Define the Godel-set for the Sheffer stroke NAND. Here the arguments 𝑥 = ⟨𝑈, 𝑉⟩ are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
Definition	df-goal 35656	Define the Godel-set of universal quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∀𝑥𝜑] = ∀𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∀𝑔𝑁𝑈 = ⟨2o, ⟨𝑁, 𝑈⟩⟩
Definition	df-sat 35657*	Define the satisfaction predicate. This recursive construction builds up a function over wff codes (see satff 35724) and simultaneously defines the set of assignments to all variables from 𝑀 that makes the coded wff true in the model 𝑀, where ∈ is interpreted as the binary relation 𝐸 on 𝑀. The interpretation of the statement 𝑆 ∈ (((𝑀 Sat 𝐸)‘𝑛)‘𝑈) is that for the model ⟨𝑀, 𝐸⟩, 𝑆:ω⟶𝑀 is a valuation of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) and 𝑈 is a code for a wff using ∈ , ⊼ , ∀ that is true under the assignment 𝑆. The function is defined by finite recursion; ((𝑀 Sat 𝐸)‘𝑛) only operates on wffs of depth at most 𝑛 ∈ ω, and ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑛 ∈ ω((𝑀 Sat 𝐸)‘𝑛) operates on all wffs. The coding scheme for the wffs is defined so that vi ∈ vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩, (𝜑 ⊼ 𝜓) is coded as ⟨1o, ⟨𝜑, 𝜓⟩⟩, and ∀ vi 𝜑 is coded as ⟨2o, ⟨𝑖, 𝜑⟩⟩. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))
Definition	df-sate 35658*	A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable 𝑛. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
Definition	df-fmla 35659	Define the predicate which defines the set of valid Godel formulas. The parameter 𝑛 defines the maximum height of the formulas: the set (Fmla‘∅) is all formulas of the form 𝑥 ∈ 𝑦 (which in our coding scheme is the set ({∅} × (ω × ω)); see df-sat 35657 for the full coding scheme), see fmla0 35696, and each extra level adds to the complexity of the formulas in (Fmla‘𝑛), see fmlasuc 35700. Remark: it is sufficient to have atomic formulas of the form 𝑥 ∈ 𝑦 only, because equations (formulas of the form 𝑥 = 𝑦), which are required as (atomic) formulas, can be introduced as a defined notion in terms of ∈𝑔, see df-goeq 35758. (Fmla‘ω) = ∪ 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas, see fmla 35695. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
Definition	df-prv 35660*	Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the empty "model" 𝑀 = ∅, since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model. Statement prv0 35744 shows that our definition yields ∅⊧𝑈 for all formulas, though of course the formula ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
Theorem	goel 35661	A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi ∈ vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
Theorem	goelel3xp 35662	A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.)
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (ω × (ω × ω)))
Theorem	goeleq12bg 35663	Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)
⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
Theorem	gonafv 35664	The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Theorem	goaleq12d 35665	Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.)
⊢ (𝜑 → 𝑀 = 𝑁)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)
Theorem	gonanegoal 35666	The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.)
⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢
Theorem	satf 35667*	The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 14-Sep-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))
Theorem	satfsucom 35668*	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at an element of the successor of ω. (Contributed by AV, 22-Sep-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
Theorem	satfn 35669	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 is a function over suc ω. (Contributed by AV, 6-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) Fn suc ω)
Theorem	satom 35670*	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at omega (ω). (Contributed by AV, 6-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑛 ∈ ω ((𝑀 Sat 𝐸)‘𝑛))
Theorem	satfvsucom 35671*	The satisfaction predicate as function over wff codes at a successor of ω. (Contributed by AV, 22-Sep-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → (𝑆‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
Theorem	satfv0 35672*	The value of the satisfaction predicate as function over wff codes at ∅. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
Theorem	satfvsuclem1 35673*	Lemma 1 for satfvsuc 35675. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀 ↑m ω))} ∈ V)
Theorem	satfvsuclem2 35674*	Lemma 2 for satfvsuc 35675. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} ∈ V)
Theorem	satfvsuc 35675*	The value of the satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 10-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
Theorem	satfv1lem 35676*	Lemma for satfv1 35677. (Contributed by AV, 9-Nov-2023.)
⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)}} = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})
Theorem	satfv1 35677*	The value of the satisfaction predicate as function over wff codes of height 1. (Contributed by AV, 9-Nov-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘1o) = ((𝑆‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
Theorem	satfsschain 35678	The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
Theorem	satfvsucsuc 35679*	The satisfaction predicate as function over wff codes of height (𝑁 + 1), expressed by the minimally necessary satisfaction predicates as function over wff codes of height 𝑁. (Contributed by AV, 21-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   ⊢ 𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))}))
Theorem	satfbrsuc 35680*	The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝑃 = (𝑆‘𝑁)    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))))
Theorem	satfrel 35681	The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
Theorem	satfdmlem 35682*	Lemma for satfdm 35683. (Contributed by AV, 12-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
Theorem	satfdm 35683*	The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
Theorem	satfrnmapom 35684	The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω))
Theorem	satfv0fun 35685	The value of the satisfaction predicate as function over wff codes at ∅ is a function. (Contributed by AV, 15-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
Theorem	satf0 35686*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation. (Contributed by AV, 14-Sep-2023.)
⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω)
Theorem	satf0sucom 35687*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.)
⊢ (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁))
Theorem	satf00 35688*	The value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at ∅. (Contributed by AV, 14-Sep-2023.)
⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
Theorem	satf0suclem 35689*	Lemma for satf0suc 35690, sat1el2xp 35693 and fmlasuc0 35698. (Contributed by AV, 19-Sep-2023.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
Theorem	satf0suc 35690*	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation at a successor. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
Theorem	satf0op 35691*	An element of a value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation expressed as ordered pair. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑋 ∈ (𝑆‘𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘𝑁))))
Theorem	satf0n0 35692	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation does not contain the empty set. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
Theorem	sat1el2xp 35693*	The first component of an element of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation is a member of a doubled Cartesian product. (Contributed by AV, 17-Sep-2023.)
⊢ (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎∃𝑏(1st ‘𝑤) ∈ (ω × (𝑎 × 𝑏)))
Theorem	fmlafv 35694	The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.)
⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
Theorem	fmla 35695	The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
Theorem	fmla0 35696*	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 14-Sep-2023.)
⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
Theorem	fmla0xp 35697	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
⊢ (Fmla‘∅) = ({∅} × (ω × ω))
Theorem	fmlasuc0 35698*	The valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 18-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
Theorem	fmlafvel 35699	A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
Theorem	fmlasuc 35700*	The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))