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Theorem fmlafvel 32918
Description: 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.)
Assertion
Ref Expression
fmlafvel (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))

Proof of Theorem fmlafvel
Dummy variables 𝑢 𝑣 𝑥 𝑦 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6674 . . . . . . 7 (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
21eleq2d 2818 . . . . . 6 (𝑥 = ∅ → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘∅)))
3 fveq2 6674 . . . . . . 7 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
43eleq2d 2818 . . . . . 6 (𝑥 = ∅ → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
52, 4bibi12d 349 . . . . 5 (𝑥 = ∅ → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅))))
65imbi2d 344 . . . 4 (𝑥 = ∅ → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))))
7 fveq2 6674 . . . . . . 7 (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
87eleq2d 2818 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑦)))
9 fveq2 6674 . . . . . . 7 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
109eleq2d 2818 . . . . . 6 (𝑥 = 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
118, 10bibi12d 349 . . . . 5 (𝑥 = 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
1211imbi2d 344 . . . 4 (𝑥 = 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))))
13 fveq2 6674 . . . . . . 7 (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
1413eleq2d 2818 . . . . . 6 (𝑥 = suc 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘suc 𝑦)))
15 fveq2 6674 . . . . . . 7 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
1615eleq2d 2818 . . . . . 6 (𝑥 = suc 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
1714, 16bibi12d 349 . . . . 5 (𝑥 = suc 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦))))
1817imbi2d 344 . . . 4 (𝑥 = suc 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
19 fveq2 6674 . . . . . . 7 (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
2019eleq2d 2818 . . . . . 6 (𝑥 = 𝑁 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑁)))
21 fveq2 6674 . . . . . . 7 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
2221eleq2d 2818 . . . . . 6 (𝑥 = 𝑁 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
2320, 22bibi12d 349 . . . . 5 (𝑥 = 𝑁 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
2423imbi2d 344 . . . 4 (𝑥 = 𝑁 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))))
25 eqeq1 2742 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝐹 = (𝑖𝑔𝑗)))
26252rexbidv 3210 . . . . . . 7 (𝑥 = 𝐹 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)))
2726elrab 3588 . . . . . 6 (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)))
28 eqidd 2739 . . . . . . . 8 ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) → ∅ = ∅)
29 simpr 488 . . . . . . . 8 ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))
3028, 29jca 515 . . . . . . 7 ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)))
31 simpr 488 . . . . . . . . 9 ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))
3231anim2i 620 . . . . . . . 8 ((𝐹 ∈ V ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)))
3332ex 416 . . . . . . 7 (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
3430, 33impbid2 229 . . . . . 6 (𝐹 ∈ V → ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
3527, 34syl5bb 286 . . . . 5 (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
36 fmla0 32915 . . . . . . 7 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
3736eleq2i 2824 . . . . . 6 (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
3837a1i 11 . . . . 5 (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}))
39 satf00 32907 . . . . . . . 8 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
4039a1i 11 . . . . . . 7 (𝐹 ∈ V → ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
4140eleq2d 2818 . . . . . 6 (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}))
42 0ex 5175 . . . . . . 7 ∅ ∈ V
43 eqeq1 2742 . . . . . . . . 9 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
4443, 26bi2anan9r 640 . . . . . . . 8 ((𝑥 = 𝐹𝑦 = ∅) → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
4544opelopabga 5388 . . . . . . 7 ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
4642, 45mpan2 691 . . . . . 6 (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
4741, 46bitrd 282 . . . . 5 (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖𝑔𝑗))))
4835, 38, 473bitr4d 314 . . . 4 (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
49 eqid 2738 . . . . . . . . . . . 12 ∅ = ∅
5049biantrur 534 . . . . . . . . . . 11 (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))))
5150bicomi 227 . . . . . . . . . 10 ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)))
5251a1i 11 . . . . . . . . 9 (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))))
53 eqeq1 2742 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
54 eqeq1 2742 . . . . . . . . . . . . . . 15 (𝑥 = 𝐹 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝐹 = ((1st𝑢)⊼𝑔(1st𝑣))))
5554rexbidv 3207 . . . . . . . . . . . . . 14 (𝑥 = 𝐹 → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣))))
56 eqeq1 2742 . . . . . . . . . . . . . . 15 (𝑥 = 𝐹 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝐹 = ∀𝑔𝑖(1st𝑢)))
5756rexbidv 3207 . . . . . . . . . . . . . 14 (𝑥 = 𝐹 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)))
5855, 57orbi12d 918 . . . . . . . . . . . . 13 (𝑥 = 𝐹 → ((∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))))
5958rexbidv 3207 . . . . . . . . . . . 12 (𝑥 = 𝐹 → (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))))
6053, 59bi2anan9r 640 . . . . . . . . . . 11 ((𝑥 = 𝐹𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)))))
6160opelopabga 5388 . . . . . . . . . 10 ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)))))
6242, 61mpan2 691 . . . . . . . . 9 (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢)))))
6359elabg 3571 . . . . . . . . 9 (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))} ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st𝑢))))
6452, 62, 633bitr4d 314 . . . . . . . 8 (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
6564adantl 485 . . . . . . 7 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
6665orbi2d 915 . . . . . 6 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
67 eqid 2738 . . . . . . . . . 10 (∅ Sat ∅) = (∅ Sat ∅)
6867satf0suc 32909 . . . . . . . . 9 (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
6968eleq2d 2818 . . . . . . . 8 (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
70 elun 4039 . . . . . . . 8 (⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
7169, 70bitrdi 290 . . . . . . 7 (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
7271ad2antrr 726 . . . . . 6 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
73 fmlasuc0 32917 . . . . . . . . 9 (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
7473eleq2d 2818 . . . . . . . 8 (𝑦 ∈ ω → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
7574ad2antrr 726 . . . . . . 7 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
76 elun 4039 . . . . . . . 8 (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
7776a1i 11 . . . . . . 7 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
78 simpr 488 . . . . . . . . 9 ((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
7978imp 410 . . . . . . . 8 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
8079orbi1d 916 . . . . . . 7 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
8175, 77, 803bitrd 308 . . . . . 6 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})))
8266, 72, 813bitr4rd 315 . . . . 5 (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
8382exp31 423 . . . 4 (𝑦 ∈ ω → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
846, 12, 18, 24, 48, 83finds 7629 . . 3 (𝑁 ∈ ω → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
8584com12 32 . 2 (𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
86 prcnel 3421 . . . . 5 𝐹 ∈ V → ¬ 𝐹 ∈ (Fmla‘𝑁))
8786adantr 484 . . . 4 ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ 𝐹 ∈ (Fmla‘𝑁))
88 opprc1 4785 . . . . . 6 𝐹 ∈ V → ⟨𝐹, ∅⟩ = ∅)
8988adantr 484 . . . . 5 ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ⟨𝐹, ∅⟩ = ∅)
90 satf0n0 32911 . . . . . . 7 (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
91 df-nel 3039 . . . . . . 7 (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
9290, 91sylib 221 . . . . . 6 (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
9392adantl 485 . . . . 5 ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
9489, 93eqneltrd 2852 . . . 4 ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
9587, 942falsed 380 . . 3 ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
9695ex 416 . 2 𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
9785, 96pm2.61i 185 1 (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  {cab 2716  wnel 3038  wrex 3054  {crab 3057  Vcvv 3398  cun 3841  c0 4211  cop 4522  {copab 5092  suc csuc 6174  cfv 6339  (class class class)co 7170  ωcom 7599  1st c1st 7712  𝑔cgoe 32866  𝑔cgna 32867  𝑔cgol 32868   Sat csat 32869  Fmlacfmla 32870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-map 8439  df-goel 32873  df-sat 32876  df-fmla 32878
This theorem is referenced by:  fmlasuc  32919
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