Theorem gdlopcfv7

Description: The value of the combined Gödel operations function when its first argument is 7. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv7.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlopcfv7.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlopcfv7	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 7o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ( Cnv2 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv7.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlopcfv7.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		7olt9o	⊢ 7o ∈ 9o
4		df-7o	⊢ 7o = suc 6o
5	3 4	suceldifsn0	⊢ 7o ∈ ( 9o ∖ { ∅ } )
6	5	a1i	⊢ ( 𝜑 → 7o ∈ ( 9o ∖ { ∅ } ) )
7	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
8	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
9		fvexd	⊢ ( 𝜑 → ( ℱ7 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ V )
10		2on	⊢ 2o ∈ On
11		7on	⊢ 7o ∈ 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		df-5o	⊢ 5o = suc 4o
19	17 18	eqsucidi	⊢ 4o ∈ 5o
20		5on	⊢ 5o ∈ On
21		df-6o	⊢ 6o = suc 5o
22	20 21	eqsucidi	⊢ 5o ∈ 6o
23		6on	⊢ 6o ∈ On
24	23 4	eqsucidi	⊢ 6o ∈ 7o
25	11 22 24	ontr1i	⊢ 5o ∈ 7o
26	11 19 25	ontr1i	⊢ 4o ∈ 7o
27	11 16 26	ontr1i	⊢ 3o ∈ 7o
28	11 13 27	ontr1i	⊢ 2o ∈ 7o
29	10 11 28	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 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 ⟩ ) ) ) ) ) ) )
30		1on	⊢ 1o ∈ On
31		df-2o	⊢ 2o = suc 1o
32	30 31	eqsucidi	⊢ 1o ∈ 2o
33	11 32 28	ontr1i	⊢ 1o ∈ 7o
34	30 11 33	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 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 ⟩ ) ) ) ) ) ) ) )
35	17 11 26	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 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 ⟩ ) ) ) ) )
36	14 11 27	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 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 ⟩ ) ) ) ) ) )
37	20 11 25	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 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 ⟩ ) ) ) )
38	23 11 24	gdlopcfvlem	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) = if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) )
39		eqcom	⊢ ( 7o = 𝑛 ↔ 𝑛 = 7o )
40	39	bilani	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → 𝑛 = 7o )
41	40	iftrued	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) = ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
42	37 38 41	3eqtrrd	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) = if ( 𝑛 = 5o , ( ℱ5 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 6o , ( ℱ6 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , if ( 𝑛 = 7o , ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) , ( ℱ8 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) ) ) ) )
43	35 36 42	3eqtr4rd	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → ( ℱ7 ‘ ⟨ 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 ⟩ ) ) ) ) ) ) )
44	29 34 43	3eqtr4rd	⊢ ( ( 𝜑 ∧ 7o = 𝑛 ) → ( ℱ7 ‘ ⟨ 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 ⟩ ) ) ) ) ) ) ) ) )
45		opeq1	⊢ ( 𝐴 = 𝑥 → ⟨ 𝐴 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
46	45	fveq2d	⊢ ( 𝐴 = 𝑥 → ( ℱ7 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
47	46	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( ℱ7 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) = ( ℱ7 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
48		opeq2	⊢ ( 𝐵 = 𝑦 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝑦 ⟩ )
49	48	fveq2d	⊢ ( 𝐵 = 𝑦 → ( ℱ7 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ7 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
50	49	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( ℱ7 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℱ7 ‘ ⟨ 𝐴 , 𝑦 ⟩ ) )
51		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 ⟩ ) ) ) ) ) ) ) ) )
52	6 7 8 9 44 47 50 51	mpt3fvot2d	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 7o , 𝐴 , 𝐵 ⟩ ) = ( ℱ7 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
53	1 2	gdlop7fv	⊢ ( 𝜑 → ( ℱ7 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ( Cnv2 ‘ 𝐵 ) ) )
54	52 53	eqtrd	⊢ ( 𝜑 → ( ℱ ‘ ⟨ 7o , 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ( Cnv2 ‘ 𝐵 ) ) )