gdlopcfv3

Description: The value of the combined Gödel operations function when its first argument is 3. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv3.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlopcfv3.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlopcfv3	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 3_o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv3.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlopcfv3.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		3olt9o	⊢ 3_o ∈ 9_o
4		df-3o	⊢ 3_o = suc 2_o
5	3 4	suceldifsn0	⊢ 3_o ∈ ( 9_o ∖ { ∅ } )
6	5	a1i	⊢ ( 𝜑 → 3_o ∈ ( 9_o ∖ { ∅ } ) )
7	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
8	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
9		fvexd	⊢ ( 𝜑 → ( ℱ₃ ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ V )
10		1on	⊢ 1_o ∈ On
11		3on	⊢ 3_o ∈ On
12		df-2o	⊢ 2_o = suc 1_o
13	10 12	eqsucidi	⊢ 1_o ∈ 2_o
14		2on	⊢ 2_o ∈ On
15	14 4	eqsucidi	⊢ 2_o ∈ 3_o
16	11 13 15	ontr1i	⊢ 1_o ∈ 3_o
17	10 11 16	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 3_o = 𝑛 ) → if ( 𝑛 = 1_o , ( ℱ₁ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2_o , ( ℱ₂ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) = if ( 𝑛 = 2_o , ( ℱ₂ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) )
18	14 11 15	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 3_o = 𝑛 ) → if ( 𝑛 = 2_o , ( ℱ₂ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) = if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) )
19		eqcom	⊢ ( 3_o = 𝑛 ↔ 𝑛 = 3_o )
20	19	bilani	⊢ ( ( 𝜑 ∧ 3_o = 𝑛 ) → 𝑛 = 3_o )
21	20	iftrued	⊢ ( ( 𝜑 ∧ 3_o = 𝑛 ) → if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) = ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
22	17 18 21	3eqtrrd	⊢ ( ( 𝜑 ∧ 3_o = 𝑛 ) → ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) = if ( 𝑛 = 1_o , ( ℱ₁ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2_o , ( ℱ₂ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) )
23		opeq1	⊢ ( 𝐴 = 𝑥 → ⟨ 𝐴 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
24	23	fveq2d	⊢ ( 𝐴 = 𝑥 → ( ℱ₃ ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
25	24	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( ℱ₃ ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
26		opeq2	⊢ ( 𝐵 = 𝑦 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝑦 ⟩ )
27	26	fveq2d	⊢ ( 𝐵 = 𝑦 → ( ℱ₃ ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ₃ ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
28	27	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( ℱ₃ ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ₃ ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
29		df-gdlopc	⊢ ℱ = ( 𝑛 ∈ ( 9_o ∖ { ∅ } ) , 𝑥 ∈ V , 𝑦 ∈ V ↦ if ( 𝑛 = 1_o , ( ℱ₁ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2_o , ( ℱ₂ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3_o , ( ℱ₃ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4_o , ( ℱ₄ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5_o , ( ℱ₅ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6_o , ( ℱ₆ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7_o , ( ℱ₇ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ₈ ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) )
30	6 7 8 9 22 25 28 29	mpt3fvot2d	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 3_o , 𝐴 , 𝐵 ⟩ ) = ( ℱ₃ ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
31	1 2	gdlop3fv	⊢ ( 𝜑 → ( ℱ₃ ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∖ 𝐵 ) )
32	30 31	eqtrd	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 3_o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∖ 𝐵 ) )

Metamath Proof Explorer

Theorem gdlopcfv3

Proof