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Metamath Proof Explorer

Theorem satfn

Description: The satisfaction predicate for wff codes in the model M and the binary relation E on M is a function over suc _om . (Contributed by AV, 6-Oct-2023)

		Ref	Expression
	Assertion	satfn	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑀 Sat 𝐸 ) Fn suc ω )

Proof

Step	Hyp	Ref	Expression
1		rdgfnon	⊢ rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝑓 ( ∃ 𝑣 ∈ 𝑓 ( 𝑥 = ( ( 1^st ‘ 𝑢 ) ⊼_𝑔 ( 1^st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑_m ω ) ∖ ( ( 2^nd ‘ 𝑢 ) ∩ ( 2^nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀_𝑔 𝑖 ( 1^st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2^nd ‘ 𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈_𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ( 𝑎 ‘ 𝑖 ) 𝐸 ( 𝑎 ‘ 𝑗 ) } ) } ) Fn On
2	1	a1i	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝑓 ( ∃ 𝑣 ∈ 𝑓 ( 𝑥 = ( ( 1^st ‘ 𝑢 ) ⊼_𝑔 ( 1^st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑_m ω ) ∖ ( ( 2^nd ‘ 𝑢 ) ∩ ( 2^nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀_𝑔 𝑖 ( 1^st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2^nd ‘ 𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈_𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ( 𝑎 ‘ 𝑖 ) 𝐸 ( 𝑎 ‘ 𝑗 ) } ) } ) Fn On )
3		ordom	⊢ Ord ω
4		ordsuc	⊢ ( Ord ω ↔ Ord suc ω )
5	3 4	mpbi	⊢ Ord suc ω
6		ordsson	⊢ ( Ord suc ω → suc ω ⊆ On )
7	5 6	mp1i	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → suc ω ⊆ On )
8	2 7	fnssresd	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝑓 ( ∃ 𝑣 ∈ 𝑓 ( 𝑥 = ( ( 1^st ‘ 𝑢 ) ⊼_𝑔 ( 1^st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑_m ω ) ∖ ( ( 2^nd ‘ 𝑢 ) ∩ ( 2^nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀_𝑔 𝑖 ( 1^st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2^nd ‘ 𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈_𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ( 𝑎 ‘ 𝑖 ) 𝐸 ( 𝑎 ‘ 𝑗 ) } ) } ) ↾ suc ω ) Fn suc ω )
9		satf	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑀 Sat 𝐸 ) = ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝑓 ( ∃ 𝑣 ∈ 𝑓 ( 𝑥 = ( ( 1^st ‘ 𝑢 ) ⊼_𝑔 ( 1^st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑_m ω ) ∖ ( ( 2^nd ‘ 𝑢 ) ∩ ( 2^nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀_𝑔 𝑖 ( 1^st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2^nd ‘ 𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈_𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ( 𝑎 ‘ 𝑖 ) 𝐸 ( 𝑎 ‘ 𝑗 ) } ) } ) ↾ suc ω ) )
10	9	fneq1d	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( 𝑀 Sat 𝐸 ) Fn suc ω ↔ ( rec ( ( 𝑓 ∈ V ↦ ( 𝑓 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝑓 ( ∃ 𝑣 ∈ 𝑓 ( 𝑥 = ( ( 1^st ‘ 𝑢 ) ⊼_𝑔 ( 1^st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑_m ω ) ∖ ( ( 2^nd ‘ 𝑢 ) ∩ ( 2^nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀_𝑔 𝑖 ( 1^st ‘ 𝑢 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2^nd ‘ 𝑢 ) } ) ) } ) ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈_𝑔 𝑗 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑_m ω ) ∣ ( 𝑎 ‘ 𝑖 ) 𝐸 ( 𝑎 ‘ 𝑗 ) } ) } ) ↾ suc ω ) Fn suc ω ) )
11	8 10	mpbird	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑀 Sat 𝐸 ) Fn suc ω )