Theorem gdlopcfv2 35972

Description: The value of the combined Gödel operations function when its first argument is 2. (Contributed by BTernaryTau, 19-Oct-2026.)

Hypotheses

Ref	Expression
gdlopcfv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gdlopcfv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
gdlopcfv2	⊢ (𝜑 → (ℱ ‘⟨2o, 𝐴, 𝐵⟩) = (𝐴 ∩ E ))

Proof of Theorem gdlopcfv2

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

Step	Hyp	Ref	Expression
1		2olt9o 35878	. . . . 5 ⊢ 2o ∈ 9o
2		df-2o 8502	. . . . 5 ⊢ 2o = suc 1o
3	1, 2	suceldifsn0 35788	. . . 4 ⊢ 2o ∈ (9o ∖ {∅})
4	3	a1i 11	. . 3 ⊢ (𝜑 → 2o ∈ (9o ∖ {∅}))
5		gdlopcfv2.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	5	elexd 3477	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
7		gdlopcfv2.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8	7	elexd 3477	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9		fvexd 6920	. . 3 ⊢ (𝜑 → (ℱ2‘⟨𝐴, 𝐵⟩) ∈ V)
10		1on 8514	. . . . 5 ⊢ 1o ∈ On
11		2on 8515	. . . . 5 ⊢ 2o ∈ On
12	10, 2	eqsucidi 35787	. . . . 5 ⊢ 1o ∈ 2o
13	10, 11, 12	gdlopcfvlem 35970	. . . 4 ⊢ ((𝜑 ∧ 2o = 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⟩))))))))
14		eqcom 2770	. . . . . 6 ⊢ (2o = 𝑛 ↔ 𝑛 = 2o)
15	14	bilani 510	. . . . 5 ⊢ ((𝜑 ∧ 2o = 𝑛) → 𝑛 = 2o)
16	15	iftrued 4493	. . . 4 ⊢ ((𝜑 ∧ 2o = 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⟩))))))) = (ℱ2‘⟨𝑥, 𝑦⟩))
17	13, 16	eqtr2d 2799	. . 3 ⊢ ((𝜑 ∧ 2o = 𝑛) → (ℱ2‘⟨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⟩)))))))))
18		opeq1 4843	. . . . 5 ⊢ (𝐴 = 𝑥 → ⟨𝐴, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
19	18	fveq2d 6909	. . . 4 ⊢ (𝐴 = 𝑥 → (ℱ2‘⟨𝐴, 𝑦⟩) = (ℱ2‘⟨𝑥, 𝑦⟩))
20	19	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝑥) → (ℱ2‘⟨𝐴, 𝑦⟩) = (ℱ2‘⟨𝑥, 𝑦⟩))
21		opeq2 4844	. . . . 5 ⊢ (𝐵 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩)
22	21	fveq2d 6909	. . . 4 ⊢ (𝐵 = 𝑦 → (ℱ2‘⟨𝐴, 𝐵⟩) = (ℱ2‘⟨𝐴, 𝑦⟩))
23	22	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝑦) → (ℱ2‘⟨𝐴, 𝐵⟩) = (ℱ2‘⟨𝐴, 𝑦⟩))
24		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⟩)))))))))
25	4, 6, 8, 9, 17, 20, 23, 24	mpt3fvot2d 7711	. 2 ⊢ (𝜑 → (ℱ ‘⟨2o, 𝐴, 𝐵⟩) = (ℱ2‘⟨𝐴, 𝐵⟩))
26	5, 7	gdlop2fv 35963	. 2 ⊢ (𝜑 → (ℱ2‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ E ))
27	25, 26	eqtrd 2798	1 ⊢ (𝜑 → (ℱ ‘⟨2o, 𝐴, 𝐵⟩) = (𝐴 ∩ E ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  ifcif 4485  {csn 4588  ⟨cop 4838  ⟨cotp 4861   E cep 5554  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-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-gdlop2 35954  df-gdlopc 35961

This theorem is used by: (None)