Theorem gdlopcfv5

Description: The value of the combined Gödel operations function when its first argument is 5. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv5.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlopcfv5.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlopcfv5	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 5o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ dom 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv5.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlopcfv5.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		5olt9o	⊢ 5o ∈ 9o
4		df-5o	⊢ 5o = suc 4o
5	3 4	suceldifsn0	⊢ 5o ∈ ( 9o ∖ { ∅ } )
6	5	a1i	⊢ ( 𝜑 → 5o ∈ ( 9o ∖ { ∅ } ) )
7	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
8	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
9		fvexd	⊢ ( 𝜑 → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ V )
10		2on	⊢ 2o ∈ On
11		5on	⊢ 5o ∈ On
12		df-3o	⊢ 3o = suc 2o
13	10 12	eqsucidi	⊢ 2o ∈ 3o
14		3on	⊢ 3o ∈ On
15		df-4o	⊢ 4o = suc 3o
16	14 15	eqsucidi	⊢ 3o ∈ 4o
17		4on	⊢ 4o ∈ On
18	17 4	eqsucidi	⊢ 4o ∈ 5o
19	11 16 18	ontr1i	⊢ 3o ∈ 5o
20	11 13 19	ontr1i	⊢ 2o ∈ 5o
21	10 11 20	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 5o = 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 ⟩ ) ) ) ) ) ) )
22		1on	⊢ 1o ∈ On
23		df-2o	⊢ 2o = suc 1o
24	22 23	eqsucidi	⊢ 1o ∈ 2o
25	11 24 20	ontr1i	⊢ 1o ∈ 5o
26	22 11 25	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 5o = 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 ⟩ ) ) ) ) ) ) ) )
27	14 11 19	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 5o = 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 ⟩ ) ) ) ) ) )
28	17 11 18	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 5o = 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 ⟩ ) ) ) ) ) = if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) )
29		eqcom	⊢ ( 5o = 𝑛 ↔ 𝑛 = 5o )
30	29	bilani	⊢ ( ( 𝜑 ∧ 5o = 𝑛 ) → 𝑛 = 5o )
31	30	iftrued	⊢ ( ( 𝜑 ∧ 5o = 𝑛 ) → if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) = ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
32	27 28 31	3eqtrrd	⊢ ( ( 𝜑 ∧ 5o = 𝑛 ) → ( ℱ5 ‘ ⟨ 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 ⟩ ) ) ) ) ) ) )
33	21 26 32	3eqtr4rd	⊢ ( ( 𝜑 ∧ 5o = 𝑛 ) → ( ℱ5 ‘ ⟨ 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 ⟩ ) ) ) ) ) ) ) ) )
34		opeq1	⊢ ( 𝐴 = 𝑥 → ⟨ 𝐴 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
35	34	fveq2d	⊢ ( 𝐴 = 𝑥 → ( ℱ5 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
36	35	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( ℱ5 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
37		opeq2	⊢ ( 𝐵 = 𝑦 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝑦 ⟩ )
38	37	fveq2d	⊢ ( 𝐵 = 𝑦 → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ5 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
39	38	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ5 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
40		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 ⟩ ) ) ) ) ) ) ) ) )
41	6 7 8 9 33 36 39 40	mpt3fvot2d	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 5o , 𝐴 , 𝐵 ⟩ ) = ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
42	1 2	gdlop5fv	⊢ ( 𝜑 → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ dom 𝐵 ) )
43	41 42	eqtrd	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 5o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ dom 𝐵 ) )