Theorem 0mpo0 7429

Description: A mapping operation with empty domain is empty. Generalization of mpo0 7431. (Contributed by AV, 27-Jan-2024.)

Assertion

Ref	Expression
0mpo0	⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐴   𝑥,𝐵   𝑦,𝐵

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem 0mpo0

Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7351	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		df-oprab 7350	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} = {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))}
3	1, 2	eqtri 2754	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))}
4		nel02 4286	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5		nel02 4286	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
6	4, 5	orim12i 908	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
7		ianor 983	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
8	6, 7	sylibr 234	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		simprl 770	. . . . . . . 8 ⊢ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
10	8, 9	nsyl 140	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
11	10	nexdv 1937	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
12	11	nexdv 1937	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
13	12	nexdv 1937	. . . 4 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
14	13	alrimiv 1928	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ∀𝑣 ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
15		eqeq1 2735	. . . . . 6 ⊢ (𝑧 = 𝑣 → (𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩))
16	15	anbi1d 631	. . . . 5 ⊢ (𝑧 = 𝑣 → ((𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))))
17	16	3exbidv 1926	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) ↔ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))))
18	17	ab0w 4326	. . 3 ⊢ ({𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))} = ∅ ↔ ∀𝑣 ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
19	14, 18	sylibr 234	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))} = ∅)
20	3, 19	eqtrid 2778	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∅c0 4280  ⟨cop 4579  {coprab 7347   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-dif 3900  df-nul 4281  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  mpo0v  7430  homffval  17596  comfffval  17604  natfval  17856  xpchomfval  18085  xpccofval  18088  plusffval  18554  efmndplusg  18788  grpsubfval  18896  grpsubfvalALT  18897  oppglsm  19554  dvrfval  20320  scaffval  20813  ipffval  21585  psrmulr  21879  marrepfval  22475  marepvfval  22480  pcofval  24937  clwwlknonmpo  30069  mendplusgfval  43284  mendmulrfval  43286  mendvscafval  43289  homf0  49120  upfval  49287