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Theorem fmlasuc 35391
Description: The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
Assertion
Ref Expression
fmlasuc (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
Distinct variable group:   𝑢,𝑁,𝑣,𝑥,𝑖

Proof of Theorem fmlasuc
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmlasuc0 35389 . 2 (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))}))
2 eqid 2737 . . . . . . . 8 (∅ Sat ∅) = (∅ Sat ∅)
32satf0op 35382 . . . . . . 7 (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4 fveq2 6906 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
54oveq2d 7447 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((1st𝑦)⊼𝑔(1st𝑧)) = ((1st𝑦)⊼𝑔(1st𝑤)))
65eqeq2d 2748 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ 𝑥 = ((1st𝑦)⊼𝑔(1st𝑤))))
76cbvrexvw 3238 . . . . . . . . 9 (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)))
87orbi1i 914 . . . . . . . 8 ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)))
9 fmlafvel 35390 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ω → (𝑧 ∈ (Fmla‘𝑁) ↔ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
109biimprd 248 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → (⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑧 ∈ (Fmla‘𝑁)))
1110adantld 490 . . . . . . . . . . . . . 14 (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑧 ∈ (Fmla‘𝑁)))
1211imp 406 . . . . . . . . . . . . 13 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑧 ∈ (Fmla‘𝑁))
13 vex 3484 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
14 0ex 5307 . . . . . . . . . . . . . . . 16 ∅ ∈ V
1513, 14op1std 8024 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, ∅⟩ → (1st𝑦) = 𝑧)
1615eleq1d 2826 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑧, ∅⟩ → ((1st𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
1716ad2antrl 728 . . . . . . . . . . . . 13 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
1812, 17mpbird 257 . . . . . . . . . . . 12 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st𝑦) ∈ (Fmla‘𝑁))
19183adant3 1133 . . . . . . . . . . 11 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))) → (1st𝑦) ∈ (Fmla‘𝑁))
20 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑢 = (1st𝑦) → (𝑢𝑔𝑣) = ((1st𝑦)⊼𝑔𝑣))
2120eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑢 = (1st𝑦) → (𝑥 = (𝑢𝑔𝑣) ↔ 𝑥 = ((1st𝑦)⊼𝑔𝑣)))
2221rexbidv 3179 . . . . . . . . . . . . 13 (𝑢 = (1st𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣)))
23 eqidd 2738 . . . . . . . . . . . . . . . 16 (𝑢 = (1st𝑦) → 𝑖 = 𝑖)
24 id 22 . . . . . . . . . . . . . . . 16 (𝑢 = (1st𝑦) → 𝑢 = (1st𝑦))
2523, 24goaleq12d 35356 . . . . . . . . . . . . . . 15 (𝑢 = (1st𝑦) → ∀𝑔𝑖𝑢 = ∀𝑔𝑖(1st𝑦))
2625eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑢 = (1st𝑦) → (𝑥 = ∀𝑔𝑖𝑢𝑥 = ∀𝑔𝑖(1st𝑦)))
2726rexbidv 3179 . . . . . . . . . . . . 13 (𝑢 = (1st𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)))
2822, 27orbi12d 919 . . . . . . . . . . . 12 (𝑢 = (1st𝑦) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))))
2928adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))) ∧ 𝑢 = (1st𝑦)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))))
302satf0op 35382 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31 fmlafvel 35390 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ω → (𝑦 ∈ (Fmla‘𝑁) ↔ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
3231biimprd 248 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ω → (⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑦 ∈ (Fmla‘𝑁)))
3332adantld 490 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑦 ∈ (Fmla‘𝑁)))
3433imp 406 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑦 ∈ (Fmla‘𝑁))
35 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
3635, 14op1std 8024 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = ⟨𝑦, ∅⟩ → (1st𝑤) = 𝑦)
3736eleq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = ⟨𝑦, ∅⟩ → ((1st𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
3837ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
3934, 38mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st𝑤) ∈ (Fmla‘𝑁))
4039adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧𝑔(1st𝑤))) → (1st𝑤) ∈ (Fmla‘𝑁))
41 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (1st𝑤) → (𝑧𝑔𝑣) = (𝑧𝑔(1st𝑤)))
4241eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (1st𝑤) → (𝑥 = (𝑧𝑔𝑣) ↔ 𝑥 = (𝑧𝑔(1st𝑤))))
4342adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧𝑔(1st𝑤))) ∧ 𝑣 = (1st𝑤)) → (𝑥 = (𝑧𝑔𝑣) ↔ 𝑥 = (𝑧𝑔(1st𝑤))))
44 simpr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧𝑔(1st𝑤))) → 𝑥 = (𝑧𝑔(1st𝑤)))
4540, 43, 44rspcedvd 3624 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧𝑔(1st𝑤))) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))
4645exp31 419 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))))
4746exlimdv 1933 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ω → (∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))))
4830, 47sylbid 240 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) → (𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))))
4948rexlimdv 3153 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣)))
5049adantr 480 . . . . . . . . . . . . . 14 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣)))
5115oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, ∅⟩ → ((1st𝑦)⊼𝑔(1st𝑤)) = (𝑧𝑔(1st𝑤)))
5251eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ↔ 𝑥 = (𝑧𝑔(1st𝑤))))
5352rexbidv 3179 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧𝑔(1st𝑤))))
5415oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑧, ∅⟩ → ((1st𝑦)⊼𝑔𝑣) = (𝑧𝑔𝑣))
5554eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔𝑣) ↔ 𝑥 = (𝑧𝑔𝑣)))
5655rexbidv 3179 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣)))
5753, 56imbi12d 344 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, ∅⟩ → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))))
5857ad2antrl 728 . . . . . . . . . . . . . 14 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧𝑔𝑣))))
5950, 58mpbird 257 . . . . . . . . . . . . 13 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣)))
6059orim1d 968 . . . . . . . . . . . 12 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))))
61603impia 1118 . . . . . . . . . . 11 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)))
6219, 29, 61rspcedvd 3624 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
63623exp 1120 . . . . . . . . 9 (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
6463exlimdv 1933 . . . . . . . 8 (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
658, 64syl7bi 255 . . . . . . 7 (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
663, 65sylbid 240 . . . . . 6 (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
6766rexlimdv 3153 . . . . 5 (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
68 fmlafvel 35390 . . . . . . . . 9 (𝑁 ∈ ω → (𝑢 ∈ (Fmla‘𝑁) ↔ ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
6968biimpa 476 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
7069adantr 480 . . . . . . 7 (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
71 vex 3484 . . . . . . . . . . . . 13 𝑢 ∈ V
7271, 14op1std 8024 . . . . . . . . . . . 12 (𝑦 = ⟨𝑢, ∅⟩ → (1st𝑦) = 𝑢)
7372oveq1d 7446 . . . . . . . . . . 11 (𝑦 = ⟨𝑢, ∅⟩ → ((1st𝑦)⊼𝑔(1st𝑧)) = (𝑢𝑔(1st𝑧)))
7473eqeq2d 2748 . . . . . . . . . 10 (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ 𝑥 = (𝑢𝑔(1st𝑧))))
7574rexbidv 3179 . . . . . . . . 9 (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧))))
76 eqidd 2738 . . . . . . . . . . . 12 (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
7776, 72goaleq12d 35356 . . . . . . . . . . 11 (𝑦 = ⟨𝑢, ∅⟩ → ∀𝑔𝑖(1st𝑦) = ∀𝑔𝑖𝑢)
7877eqeq2d 2748 . . . . . . . . . 10 (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀𝑔𝑖(1st𝑦) ↔ 𝑥 = ∀𝑔𝑖𝑢))
7978rexbidv 3179 . . . . . . . . 9 (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
8075, 79orbi12d 919 . . . . . . . 8 (𝑦 = ⟨𝑢, ∅⟩ → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
8180adantl 481 . . . . . . 7 ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) ∧ 𝑦 = ⟨𝑢, ∅⟩) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
82 fmlafvel 35390 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) ↔ ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
8382biimpd 229 . . . . . . . . . . . . . 14 (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
8483adantr 480 . . . . . . . . . . . . 13 ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
8584imp 406 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
8685adantr 480 . . . . . . . . . . 11 ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢𝑔𝑣)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
87 vex 3484 . . . . . . . . . . . . . . 15 𝑣 ∈ V
8887, 14op1std 8024 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑣, ∅⟩ → (1st𝑧) = 𝑣)
8988oveq2d 7447 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑣, ∅⟩ → (𝑢𝑔(1st𝑧)) = (𝑢𝑔𝑣))
9089eqeq2d 2748 . . . . . . . . . . . 12 (𝑧 = ⟨𝑣, ∅⟩ → (𝑥 = (𝑢𝑔(1st𝑧)) ↔ 𝑥 = (𝑢𝑔𝑣)))
9190adantl 481 . . . . . . . . . . 11 (((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢𝑔𝑣)) ∧ 𝑧 = ⟨𝑣, ∅⟩) → (𝑥 = (𝑢𝑔(1st𝑧)) ↔ 𝑥 = (𝑢𝑔𝑣)))
92 simpr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢𝑔𝑣)) → 𝑥 = (𝑢𝑔𝑣))
9386, 91, 92rspcedvd 3624 . . . . . . . . . 10 ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢𝑔𝑣)) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧)))
9493rexlimdva2 3157 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧))))
9594orim1d 968 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
9695imp 406 . . . . . . 7 (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
9770, 81, 96rspcedvd 3624 . . . . . 6 (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)))
9897rexlimdva2 3157 . . . . 5 (𝑁 ∈ ω → (∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))))
9967, 98impbid 212 . . . 4 (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)) ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
10099abbidv 2808 . . 3 (𝑁 ∈ ω → {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))} = {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})
101100uneq2d 4168 . 2 (𝑁 ∈ ω → ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦))}) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
1021, 101eqtrd 2777 1 (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  cun 3949  c0 4333  cop 4632  suc csuc 6386  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  𝑔cgna 35339  𝑔cgol 35340   Sat csat 35341  Fmlacfmla 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-goel 35345  df-goal 35347  df-sat 35348  df-fmla 35350
This theorem is referenced by:  fmla1  35392  isfmlasuc  35393  fmlasssuc  35394  fmlaomn0  35395
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