Theorem 0mpo0 7516

Description: A mapping operation with empty domain is empty. Generalization of mpo0 7518. (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 7436	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		df-oprab 7435	. . 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 2765	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))}
4		nel02 4339	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5		nel02 4339	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
6	4, 5	orim12i 909	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
7		ianor 984	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
8	6, 7	sylibr 234	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		simprl 771	. . . . . . . 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 1936	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
12	11	nexdv 1936	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
13	12	nexdv 1936	. . . 4 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
14	13	alrimiv 1927	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ∀𝑣 ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
15		eqeq1 2741	. . . . . 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 1925	. . . 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 4379	. . 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 2789	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∅c0 4333  ⟨cop 4632  {coprab 7432   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-nul 4334  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  mpo0v  7517  homffval  17733  comfffval  17741  natfval  17994  xpchomfval  18224  xpccofval  18227  plusffval  18659  efmndplusg  18893  grpsubfval  19001  grpsubfvalALT  19002  oppglsm  19660  dvrfval  20402  scaffval  20878  ipffval  21666  psrmulr  21962  marrepfval  22566  marepvfval  22571  pcofval  25043  clwwlknonmpo  30108  mendplusgfval  43193  mendmulrfval  43195  mendvscafval  43198  upfval  48933