Theorem 0mpo0 7510

Description: A mapping operation with empty domain is empty. Generalization of mpo0 7512. (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 7431	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		df-oprab 7430	. . 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 4335	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5		nel02 4335	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
6	4, 5	orim12i 906	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
7		ianor 979	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
8	6, 7	sylibr 233	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		simprl 769	. . . . . . . 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 1932	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
12	11	nexdv 1932	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
13	12	nexdv 1932	. . . 4 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
14	13	alrimiv 1923	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → ∀𝑣 ¬ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
15		eqeq1 2730	. . . . . 6 ⊢ (𝑧 = 𝑣 → (𝑧 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩))
16	15	anbi1d 629	. . . . 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 1921	. . . 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 4378	. . 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 233	. 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 394   ∨ wo 845  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2703  ∅c0 4325  ⟨cop 4639  {coprab 7427   ∈ cmpo 7428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-dif 3950  df-nul 4326  df-oprab 7430  df-mpo 7431

This theorem is referenced by:  mpo0v  7511  homffval  17705  comfffval  17713  natfval  17971  xpchomfval  18205  xpccofval  18208  plusffval  18641  efmndplusg  18872  grpsubfval  18980  grpsubfvalALT  18981  oppglsm  19642  dvrfval  20386  scaffval  20858  ipffval  21646  psrmulr  21953  marrepfval  22556  marepvfval  22561  pcofval  25031  clwwlknonmpo  30025  mendplusgfval  42864  mendmulrfval  42866  mendvscafval  42869