Theorem gdlopcfv4

Description: The value of the combined Gödel operations function when its first argument is 4. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv4.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlopcfv4.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlopcfv4	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 4o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv4.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlopcfv4.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		4olt9o	⊢ 4o ∈ 9o
4		df-4o	⊢ 4o = suc 3o
5	3 4	suceldifsn0	⊢ 4o ∈ ( 9o ∖ { ∅ } )
6	5	a1i	⊢ ( 𝜑 → 4o ∈ ( 9o ∖ { ∅ } ) )
7	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
8	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
9		fvexd	⊢ ( 𝜑 → ( ℱ4 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ V )
10		2on	⊢ 2o ∈ On
11		4on	⊢ 4o ∈ On
12		df-3o	⊢ 3o = suc 2o
13	10 12	eqsucidi	⊢ 2o ∈ 3o
14		3on	⊢ 3o ∈ On
15	14 4	eqsucidi	⊢ 3o ∈ 4o
16	11 13 15	ontr1i	⊢ 2o ∈ 4o
17	10 11 16	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) = if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) )
18	14 11 15	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) = if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) )
19		eqcom	⊢ ( 4o = 𝑛 ↔ 𝑛 = 4o )
20	19	bilani	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → 𝑛 = 4o )
21	20	iftrued	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) = ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
22	17 18 21	3eqtrrd	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) = if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) )
23		1on	⊢ 1o ∈ On
24		df-2o	⊢ 2o = suc 1o
25	23 24	eqsucidi	⊢ 1o ∈ 2o
26	14 25 13	ontr1i	⊢ 1o ∈ 3o
27	11 26 15	ontr1i	⊢ 1o ∈ 4o
28	23 11 27	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → if ( 𝑛 = 1o , ( ℱ1 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) = if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) )
29	22 28	eqtr4d	⊢ ( ( 𝜑 ∧ 4o = 𝑛 ) → ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) = if ( 𝑛 = 1o , ( ℱ1 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) )
30		opeq1	⊢ ( 𝐴 = 𝑥 → ⟨ 𝐴 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
31	30	fveq2d	⊢ ( 𝐴 = 𝑥 → ( ℱ4 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
32	31	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( ℱ4 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
33		opeq2	⊢ ( 𝐵 = 𝑦 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝑦 ⟩ )
34	33	fveq2d	⊢ ( 𝐵 = 𝑦 → ( ℱ4 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ4 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
35	34	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( ℱ4 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ4 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
36		df-gdlopc	⊢ ℱ = ( 𝑛 ∈ ( 9o ∖ { ∅ } ) , 𝑥 ∈ V , 𝑦 ∈ V ↦ if ( 𝑛 = 1o , ( ℱ1 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 2o , ( ℱ2 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 3o , ( ℱ3 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 4o , ( ℱ4 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) ) ) ) ) )
37	6 7 8 9 29 32 35 36	mpt3fvot2d	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 4o , 𝐴 , 𝐵 ⟩ ) = ( ℱ4 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
38	1 2	gdlop4fv	⊢ ( 𝜑 → ( ℱ4 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ↾ 𝐵 ) )
39	37 38	eqtrd	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 4o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ↾ 𝐵 ) )