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Theorem fmlasuc 35371
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 35369 . 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 2735 . . . . . . . 8 (∅ Sat ∅) = (∅ Sat ∅)
32satf0op 35362 . . . . . . 7 (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
54oveq2d 7447 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((1st𝑦)⊼𝑔(1st𝑧)) = ((1st𝑦)⊼𝑔(1st𝑤)))
65eqeq2d 2746 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ 𝑥 = ((1st𝑦)⊼𝑔(1st𝑤))))
76cbvrexvw 3236 . . . . . . . . 9 (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)))
87orbi1i 913 . . . . . . . 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 35370 . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
14 0ex 5313 . . . . . . . . . . . . . . . 16 ∅ ∈ V
1513, 14op1std 8023 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, ∅⟩ → (1st𝑦) = 𝑧)
1615eleq1d 2824 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . 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 2746 . . . . . . . . . . . . . 14 (𝑢 = (1st𝑦) → (𝑥 = (𝑢𝑔𝑣) ↔ 𝑥 = ((1st𝑦)⊼𝑔𝑣)))
2221rexbidv 3177 . . . . . . . . . . . . 13 (𝑢 = (1st𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st𝑦)⊼𝑔𝑣)))
23 eqidd 2736 . . . . . . . . . . . . . . . 16 (𝑢 = (1st𝑦) → 𝑖 = 𝑖)
24 id 22 . . . . . . . . . . . . . . . 16 (𝑢 = (1st𝑦) → 𝑢 = (1st𝑦))
2523, 24goaleq12d 35336 . . . . . . . . . . . . . . 15 (𝑢 = (1st𝑦) → ∀𝑔𝑖𝑢 = ∀𝑔𝑖(1st𝑦))
2625eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑢 = (1st𝑦) → (𝑥 = ∀𝑔𝑖𝑢𝑥 = ∀𝑔𝑖(1st𝑦)))
2726rexbidv 3177 . . . . . . . . . . . . 13 (𝑢 = (1st𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦)))
2822, 27orbi12d 918 . . . . . . . . . . . 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 35362 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31 fmlafvel 35370 . . . . . . . . . . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 ∈ V
3635, 14op1std 8023 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = ⟨𝑦, ∅⟩ → (1st𝑤) = 𝑦)
3736eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . . . . . 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 1931 . . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑤)) ↔ 𝑥 = (𝑧𝑔(1st𝑤))))
5352rexbidv 3177 . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔𝑣) ↔ 𝑥 = (𝑧𝑔𝑣)))
5655rexbidv 3177 . . . . . . . . . . . . . . . 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 967 . . . . . . . . . . . 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 1116 . . . . . . . . . . 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 1118 . . . . . . . . 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 1931 . . . . . . . 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 3151 . . . . 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 35370 . . . . . . . . 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 3482 . . . . . . . . . . . . 13 𝑢 ∈ V
7271, 14op1std 8023 . . . . . . . . . . . 12 (𝑦 = ⟨𝑢, ∅⟩ → (1st𝑦) = 𝑢)
7372oveq1d 7446 . . . . . . . . . . 11 (𝑦 = ⟨𝑢, ∅⟩ → ((1st𝑦)⊼𝑔(1st𝑧)) = (𝑢𝑔(1st𝑧)))
7473eqeq2d 2746 . . . . . . . . . 10 (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ 𝑥 = (𝑢𝑔(1st𝑧))))
7574rexbidv 3177 . . . . . . . . 9 (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑦)⊼𝑔(1st𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧))))
76 eqidd 2736 . . . . . . . . . . . 12 (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
7776, 72goaleq12d 35336 . . . . . . . . . . 11 (𝑦 = ⟨𝑢, ∅⟩ → ∀𝑔𝑖(1st𝑦) = ∀𝑔𝑖𝑢)
7877eqeq2d 2746 . . . . . . . . . 10 (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀𝑔𝑖(1st𝑦) ↔ 𝑥 = ∀𝑔𝑖𝑢))
7978rexbidv 3177 . . . . . . . . 9 (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
8075, 79orbi12d 918 . . . . . . . 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 35370 . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . 15 𝑣 ∈ V
8887, 14op1std 8023 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑣, ∅⟩ → (1st𝑧) = 𝑣)
8988oveq2d 7447 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑣, ∅⟩ → (𝑢𝑔(1st𝑧)) = (𝑢𝑔𝑣))
9089eqeq2d 2746 . . . . . . . . . . . 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 3155 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢𝑔(1st𝑧))))
9594orim1d 967 . . . . . . . 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 3155 . . . . 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 2806 . . 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 4178 . 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 2775 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 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  cun 3961  c0 4339  cop 4637  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  𝑔cgna 35319  𝑔cgol 35320   Sat csat 35321  Fmlacfmla 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-goel 35325  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  fmla1  35372  isfmlasuc  35373  fmlasssuc  35374  fmlaomn0  35375
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