Theorem gdlopcfv6

Description: The value of the combined Gödel operations function when its first argument is 6. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv6.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlopcfv6.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlopcfv6	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 6o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ◡ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv6.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlopcfv6.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		6olt9o	⊢ 6o ∈ 9o
4		df-6o	⊢ 6o = suc 5o
5	3 4	suceldifsn0	⊢ 6o ∈ ( 9o ∖ { ∅ } )
6	5	a1i	⊢ ( 𝜑 → 6o ∈ ( 9o ∖ { ∅ } ) )
7	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
8	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
9		fvexd	⊢ ( 𝜑 → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ V )
10		3on	⊢ 3o ∈ On
11		6on	⊢ 6o ∈ On
12		df-4o	⊢ 4o = suc 3o
13	10 12	eqsucidi	⊢ 3o ∈ 4o
14		4on	⊢ 4o ∈ On
15		df-5o	⊢ 5o = suc 4o
16	14 15	eqsucidi	⊢ 4o ∈ 5o
17		5on	⊢ 5o ∈ On
18	17 4	eqsucidi	⊢ 5o ∈ 6o
19	11 16 18	ontr1i	⊢ 4o ∈ 6o
20	11 13 19	ontr1i	⊢ 3o ∈ 6o
21	10 11 20	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 6o = 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 ⟩ ) ) ) ) ) )
22		2on	⊢ 2o ∈ On
23		df-3o	⊢ 3o = suc 2o
24	22 23	eqsucidi	⊢ 2o ∈ 3o
25	11 24 20	ontr1i	⊢ 2o ∈ 6o
26	22 11 25	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 6o = 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 ⟩ ) ) ) ) ) ) )
27	14 11 19	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 6o = 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 ⟩ ) ) ) ) )
28	17 11 18	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 6o = 𝑛 ) → if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) = if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) )
29		eqcom	⊢ ( 6o = 𝑛 ↔ 𝑛 = 6o )
30	29	bilani	⊢ ( ( 𝜑 ∧ 6o = 𝑛 ) → 𝑛 = 6o )
31	30	iftrued	⊢ ( ( 𝜑 ∧ 6o = 𝑛 ) → if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) = ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
32	27 28 31	3eqtrrd	⊢ ( ( 𝜑 ∧ 6o = 𝑛 ) → ( ℱ6 ‘ ⟨ 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 ∧ 6o = 𝑛 ) → ( ℱ6 ‘ ⟨ 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		1on	⊢ 1o ∈ On
35		df-2o	⊢ 2o = suc 1o
36	34 35	eqsucidi	⊢ 1o ∈ 2o
37	11 36 25	ontr1i	⊢ 1o ∈ 6o
38	34 11 37	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 6o = 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 ⟩ ) ) ) ) ) ) ) )
39	33 38	eqtr4d	⊢ ( ( 𝜑 ∧ 6o = 𝑛 ) → ( ℱ6 ‘ ⟨ 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 ⟩ ) ) ) ) ) ) ) ) )
40		opeq1	⊢ ( 𝐴 = 𝑥 → ⟨ 𝐴 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
41	40	fveq2d	⊢ ( 𝐴 = 𝑥 → ( ℱ6 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
42	41	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( ℱ6 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
43		opeq2	⊢ ( 𝐵 = 𝑦 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝑦 ⟩ )
44	43	fveq2d	⊢ ( 𝐵 = 𝑦 → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ6 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
45	44	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ6 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
46		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 ⟩ ) ) ) ) ) ) ) ) )
47	6 7 8 9 39 42 45 46	mpt3fvot2d	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 6o , 𝐴 , 𝐵 ⟩ ) = ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
48	1 2	gdlop6fv	⊢ ( 𝜑 → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ◡ 𝐵 ) )
49	47 48	eqtrd	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 6o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ◡ 𝐵 ) )