Theorem gdlop3fv 35964

Description: The value of the third Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)

Hypotheses

Ref	Expression
gdlop3fv.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gdlop3fv.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
gdlop3fv	⊢ (𝜑 → (ℱ3‘⟨𝐴, 𝐵⟩) = (𝐴 ∖ 𝐵))

Proof of Theorem gdlop3fv

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

Step	Hyp	Ref	Expression
1		gdlop3fv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	elexd 3477	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		gdlop3fv.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3477	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5	1	difexd 5297	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
6		difeq1 4059	. . 3 ⊢ (𝐴 = 𝑥 → (𝐴 ∖ 𝑦) = (𝑥 ∖ 𝑦))
7	6	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝑥) → (𝐴 ∖ 𝑦) = (𝑥 ∖ 𝑦))
8		difeq2 4060	. . 3 ⊢ (𝐵 = 𝑦 → (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝑦))
9	8	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝑦) → (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝑦))
10		df-gdlop3 35955	. 2 ⊢ ℱ3 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∖ 𝑦))
11	2, 4, 5, 7, 9, 10	mpofvop2d 35796	1 ⊢ (𝜑 → (ℱ3‘⟨𝐴, 𝐵⟩) = (𝐴 ∖ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  ⟨cop 4838  ‘cfv 6555  ℱ₃cgdlop3 35944

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-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-oprab 7444  df-mpo 7445  df-gdlop3 35955

This theorem is used by:  gdlopcfv3  35973