Theorem 0mpo0 7237

Description: A mapping operation with empty domain is empty. Generalization of mpo0 7239. (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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7161	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		df-oprab 7160	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
3	1, 2	eqtri 2844	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
4		nel02 4298	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5		nel02 4298	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
6	4, 5	orim12i 905	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
7		ianor 978	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
8	6, 7	sylibr 236	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		simprl 769	. . . . . . . 8 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
10	8, 9	nsyl 142	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
11	10	nexdv 1937	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
12	11	nexdv 1937	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
13	12	nexdv 1937	. . . 4 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
14	13	alrimiv 1928	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ∀𝑤 ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
15		ab0 4333	. . 3 ⊢ ({𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅ ↔ ∀𝑤 ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
16	14, 15	sylibr 236	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅)
17	3, 16	syl5eq 2868	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  ∅c0 4291  ⟨cop 4573  {coprab 7157   ∈ cmpo 7158

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-dif 3939  df-nul 4292  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  mpo0v  7238  homffval  16960  comfffval  16968  natfval  17216  xpchomfval  17429  xpccofval  17432  plusffval  17858  efmndplusg  18045  grpsubfval  18147  grpsubfvalALT  18148  oppglsm  18767  dvrfval  19434  scaffval  19652  psrmulr  20164  ipffval  20792  marrepfval  21169  marepvfval  21174  pcofval  23614  clwwlknonmpo  27868  mendplusgfval  39834  mendmulrfval  39836  mendvscafval  39839