Metamath Proof Explorer


Theorem satfvsucom

Description: The satisfaction predicate as function over wff codes at a successor of _om . (Contributed by AV, 22-Sep-2023)

Ref Expression
Hypothesis satfvsucom.s 𝑆 = ( 𝑀 Sat 𝐸 )
Assertion satfvsucom ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → ( 𝑆𝑁 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 satfvsucom.s 𝑆 = ( 𝑀 Sat 𝐸 )
2 satf ( ( 𝑀𝑉𝐸𝑊 ) → ( 𝑀 Sat 𝐸 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) )
3 2 3adant3 ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → ( 𝑀 Sat 𝐸 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) )
4 1 3 syl5eq ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → 𝑆 = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) )
5 4 fveq1d ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → ( 𝑆𝑁 ) = ( ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) ‘ 𝑁 ) )
6 fvres ( 𝑁 ∈ suc ω → ( ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) ‘ 𝑁 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ‘ 𝑁 ) )
7 6 3ad2ant3 ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → ( ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ↾ suc ω ) ‘ 𝑁 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ‘ 𝑁 ) )
8 5 7 eqtrd ( ( 𝑀𝑉𝐸𝑊𝑁 ∈ suc ω ) → ( 𝑆𝑁 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢𝑓 ( ∃ 𝑣𝑓 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ∀ 𝑧𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎𝑖 ) 𝐸 ( 𝑎𝑗 ) } ) } ) ‘ 𝑁 ) )