Theorem eldm3 35783

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 3485	. 2 ⊢ (𝐴 ∈ dom 𝐵 → 𝐴 ∈ V)
2		snprc 4698	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		reseq2 5966	. . . . 5 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4		res0 5975	. . . . 5 ⊢ (𝐵 ↾ ∅) = ∅
5	3, 4	eqtrdi 2787	. . . 4 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
6	2, 5	sylbi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
7	6	necon1ai 2960	. 2 ⊢ ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2823	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵 ↔ 𝐴 ∈ dom 𝐵))
9		sneq 4616	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	reseq2d 5971	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
11	10	neeq1d 2992	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12		dfclel 2811	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
13	12	exbii 1848	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
14		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
15	14	eldm2 5886	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
16		n0 4333	. . . . 5 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17		elres 6012	. . . . . . 7 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18		eleq1 2823	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
19	18	pm5.32i 574	. . . . . . . . . 10 ⊢ ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20		opeq1 4854	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
21	20	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
22	21	anbi1d 631	. . . . . . . . . 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 1921	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
25	14, 24	rexsn 4663	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
26	17, 25	bitri 275	. . . . . 6 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
27	26	exbii 1848	. . . . 5 ⊢ (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
28		excom 2163	. . . . 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 3541	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
32	1, 7, 31	pm5.21nii 378	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  Vcvv 3464  ∅c0 4313  {csn 4606  ⟨cop 4612  dom cdm 5659   ↾ cres 5661

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-dm 5669  df-res 5671

This theorem is referenced by:  elrn3  35784