Theorem gdlopcfv4 35974

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

Hypotheses

Ref	Expression
gdlopcfv4.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gdlopcfv4.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
gdlopcfv4	⊢ (𝜑 → (ℱ ‘⟨4o, 𝐴, 𝐵⟩) = (𝐴 ↾ 𝐵))

Proof of Theorem gdlopcfv4

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		4olt9o 35876	. . . . 5 ⊢ 4o ∈ 9o
2		df-4o 8504	. . . . 5 ⊢ 4o = suc 3o
3	1, 2	suceldifsn0 35788	. . . 4 ⊢ 4o ∈ (9o ∖ {∅})
4	3	a1i 11	. . 3 ⊢ (𝜑 → 4o ∈ (9o ∖ {∅}))
5		gdlopcfv4.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	5	elexd 3477	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
7		gdlopcfv4.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8	7	elexd 3477	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9		fvexd 6920	. . 3 ⊢ (𝜑 → (ℱ4‘⟨𝐴, 𝐵⟩) ∈ V)
10		2on 8515	. . . . . 6 ⊢ 2o ∈ On
11		4on 8519	. . . . . 6 ⊢ 4o ∈ On
12		df-3o 8503	. . . . . . . 8 ⊢ 3o = suc 2o
13	10, 12	eqsucidi 35787	. . . . . . 7 ⊢ 2o ∈ 3o
14		3on 8518	. . . . . . . 8 ⊢ 3o ∈ On
15	14, 2	eqsucidi 35787	. . . . . . 7 ⊢ 3o ∈ 4o
16	11, 13, 15	ontr1i 35786	. . . . . 6 ⊢ 2o ∈ 4o
17	10, 11, 16	gdlopcfvlem 35970	. . . . 5 ⊢ ((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 35970	. . . . 5 ⊢ ((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 2770	. . . . . . 7 ⊢ (4o = 𝑛 ↔ 𝑛 = 4o)
20	19	bilani 510	. . . . . 6 ⊢ ((𝜑 ∧ 4o = 𝑛) → 𝑛 = 4o)
21	20	iftrued 4493	. . . . 5 ⊢ ((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 2803	. . . 4 ⊢ ((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 8514	. . . . 5 ⊢ 1o ∈ On
24		df-2o 8502	. . . . . . . 8 ⊢ 2o = suc 1o
25	23, 24	eqsucidi 35787	. . . . . . 7 ⊢ 1o ∈ 2o
26	14, 25, 13	ontr1i 35786	. . . . . 6 ⊢ 1o ∈ 3o
27	11, 26, 15	ontr1i 35786	. . . . 5 ⊢ 1o ∈ 4o
28	23, 11, 27	gdlopcfvlem 35970	. . . 4 ⊢ ((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 2801	. . 3 ⊢ ((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 4843	. . . . 5 ⊢ (𝐴 = 𝑥 → ⟨𝐴, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
31	30	fveq2d 6909	. . . 4 ⊢ (𝐴 = 𝑥 → (ℱ4‘⟨𝐴, 𝑦⟩) = (ℱ4‘⟨𝑥, 𝑦⟩))
32	31	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝑥) → (ℱ4‘⟨𝐴, 𝑦⟩) = (ℱ4‘⟨𝑥, 𝑦⟩))
33		opeq2 4844	. . . . 5 ⊢ (𝐵 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩)
34	33	fveq2d 6909	. . . 4 ⊢ (𝐵 = 𝑦 → (ℱ4‘⟨𝐴, 𝐵⟩) = (ℱ4‘⟨𝐴, 𝑦⟩))
35	34	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝑦) → (ℱ4‘⟨𝐴, 𝐵⟩) = (ℱ4‘⟨𝐴, 𝑦⟩))
36		df-gdlopc 35961	. . 3 ⊢ 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	4, 6, 8, 9, 29, 32, 35, 36	mpt3fvot2d 7711	. 2 ⊢ (𝜑 → (ℱ ‘⟨4o, 𝐴, 𝐵⟩) = (ℱ4‘⟨𝐴, 𝐵⟩))
38	5, 7	gdlop4fv 35965	. 2 ⊢ (𝜑 → (ℱ4‘⟨𝐴, 𝐵⟩) = (𝐴 ↾ 𝐵))
39	37, 38	eqtrd 2798	1 ⊢ (𝜑 → (ℱ ‘⟨4o, 𝐴, 𝐵⟩) = (𝐴 ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  ifcif 4485  {csn 4588  ⟨cop 4838  ⟨cotp 4861   ↾ cres 5657  Oncon0 6377  ‘cfv 6555  1_oc1o 8494  2_oc2o 8495  3_oc3o 8496  4_oc4o 8497  5_oc5o 35852  6_oc6o 35853  7_oc7o 35854  9_oc9o 35856  ℱ₁cgdlop1 35942  ℱ₂cgdlop2 35943  ℱ₃cgdlop3 35944  ℱ₄cgdlop4 35945  ℱ₅cgdlop5 35946  ℱ₆cgdlop6 35947  ℱ₇cgdlop7 35948  ℱ₈cgdlop8 35949  ℱ cgdlopc 35950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fv 6563  df-oprab 7444  df-mpo 7445  df-mpt3 7705  df-1o 8501  df-2o 8502  df-3o 8503  df-4o 8504  df-5o 35857  df-6o 35858  df-7o 35859  df-8o 35860  df-9o 35861  df-gdlop4 35956  df-gdlopc 35961

This theorem is used by: (None)