Theorem fvtp0 7237

Description: The undefined value of a function with a domain of three elements. (Contributed by AV, 18-Aug-2026.)

Hypotheses

Ref	Expression
fvtp0.d	⊢ 𝐷 ∈ V
fvtp0.e	⊢ 𝐸 ∈ V
fvtp0.f	⊢ 𝐹 ∈ V
fvtp0.x	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
fvtp0	⊢ ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝑋) = ∅)

Proof of Theorem fvtp0

Step	Hyp	Ref	Expression
1		df-ne 2960	. . . . 5 ⊢ (𝑋 ≠ 𝐴 ↔ ¬ 𝑋 = 𝐴)
2		df-ne 2960	. . . . 5 ⊢ (𝑋 ≠ 𝐵 ↔ ¬ 𝑋 = 𝐵)
3		df-ne 2960	. . . . 5 ⊢ (𝑋 ≠ 𝐶 ↔ ¬ 𝑋 = 𝐶)
4	1, 2, 3	3anbi123i 1173	. . . 4 ⊢ ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) ↔ (¬ 𝑋 = 𝐴 ∧ ¬ 𝑋 = 𝐵 ∧ ¬ 𝑋 = 𝐶))
5		3ioran 1123	. . . . 5 ⊢ (¬ (𝑋 = 𝐴 ∨ 𝑋 = 𝐵 ∨ 𝑋 = 𝐶) ↔ (¬ 𝑋 = 𝐴 ∧ ¬ 𝑋 = 𝐵 ∧ ¬ 𝑋 = 𝐶))
6		fvtp0.x	. . . . . 6 ⊢ 𝑋 ∈ V
7	6	eltp 4651	. . . . 5 ⊢ (𝑋 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑋 = 𝐴 ∨ 𝑋 = 𝐵 ∨ 𝑋 = 𝐶))
8	5, 7	xchnxbir 336	. . . 4 ⊢ (¬ 𝑋 ∈ {𝐴, 𝐵, 𝐶} ↔ (¬ 𝑋 = 𝐴 ∧ ¬ 𝑋 = 𝐵 ∧ ¬ 𝑋 = 𝐶))
9	4, 8	sylbb2 241	. . 3 ⊢ ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) → ¬ 𝑋 ∈ {𝐴, 𝐵, 𝐶})
10		fvtp0.d	. . . . 5 ⊢ 𝐷 ∈ V
11		fvtp0.e	. . . . 5 ⊢ 𝐸 ∈ V
12		fvtp0.f	. . . . 5 ⊢ 𝐹 ∈ V
13	10, 11, 12	dmtpop 6238	. . . 4 ⊢ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}
14	13	eleq2i 2856	. . 3 ⊢ (𝑋 ∈ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ↔ 𝑋 ∈ {𝐴, 𝐵, 𝐶})
15	9, 14	sylnibr 332	. 2 ⊢ ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) → ¬ 𝑋 ∈ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
16		ndmfv 6948	. 2 ⊢ (¬ 𝑋 ∈ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝑋) = ∅)
17	15, 16	syl 18	1 ⊢ ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝑋) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {ctp 4592  ⟨cop 4840  dom cdm 5658  ‘cfv 6566

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-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  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-tp 4593  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574

This theorem is used by:  degenmgm  19187  degenmgm2  19190