Theorem 0mpo0 7440

Description: A mapping operation with empty domain is empty. Generalization of mpo0 7442. (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 7362	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		df-oprab 7361	. . 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 2762	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))}
4		nel02 4268	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5		nel02 4268	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
6	4, 5	orim12i 914	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
7		ianor 989	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
8	6, 7	sylibr 235	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		simprl 776	. . . . . . . 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 1943	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
12	11	nexdv 1943	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
13	12	nexdv 1943	. . . 4 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
14	13	alrimiv 1934	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ∀𝑣 ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
15		eqeq1 2743	. . . . . 6 ⊢ (𝑧 = 𝑣 → (𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩))
16	15	anbi1d 637	. . . . 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 1932	. . . 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 4308	. . 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 235	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → {𝑧 ∣ ∃𝑥∃𝑦∃𝑤(𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶))} = ∅)
20	3, 19	eqtrid 2786	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∅c0 4262  ⟨cop 4562  {coprab 7358   ∈ cmpo 7359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-dif 3886  df-nul 4263  df-oprab 7361  df-mpo 7362

This theorem is referenced by:  mpo0v  7441  homffval  17648  comfffval  17656  natfval  17908  xpchomfval  18137  xpccofval  18140  plusffval  18606  efmndplusg  18840  grpsubfval  18951  grpsubfvalALT  18952  oppglsm  19609  dvrfval  20374  scaffval  20871  ipffval  21624  psrmulr  21918  marrepfval  22544  marepvfval  22549  pcofval  24996  clwwlknonmpo  30178  mendplusgfval  43635  mendmulrfval  43637  mendvscafval  43640  homf0  49507  upfval  49674