Theorem eldm3 33728

Description: Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Assertion

Ref	Expression
eldm3	⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Proof of Theorem eldm3

Dummy variables 𝑥 𝑦 𝑧 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ dom 𝐵 → 𝐴 ∈ V)
2		snprc 4653	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		reseq2 5886	. . . . 5 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4		res0 5895	. . . . 5 ⊢ (𝐵 ↾ ∅) = ∅
5	3, 4	eqtrdi 2794	. . . 4 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
6	2, 5	sylbi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
7	6	necon1ai 2971	. 2 ⊢ ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2826	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵 ↔ 𝐴 ∈ dom 𝐵))
9		sneq 4571	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	reseq2d 5891	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
11	10	neeq1d 3003	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12		dfclel 2817	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
13	12	exbii 1850	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
14		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
15	14	eldm2 5810	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
16		n0 4280	. . . . 5 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17		elres 5930	. . . . . . 7 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
19	18	pm5.32i 575	. . . . . . . . . 10 ⊢ ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20		opeq1 4804	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
21	20	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
22	21	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
23	19, 22	bitr3id 285	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
24	23	exbidv 1924	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
25	14, 24	rexsn 4618	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
26	17, 25	bitri 274	. . . . . 6 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
27	26	exbii 1850	. . . . 5 ⊢ (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
28		excom 2162	. . . . 5 ⊢ (∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
29	16, 27, 28	3bitri 297	. . . 4 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
30	13, 15, 29	3bitr4i 303	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
31	8, 11, 30	vtoclbg 3507	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
32	1, 7, 31	pm5.21nii 380	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  Vcvv 3432  ∅c0 4256  {csn 4561  ⟨cop 4567  dom cdm 5589   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-res 5601

This theorem is referenced by:  elrn3  33729