Theorem eldm3 36148

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 3484	. 2 ⊢ (𝐴 ∈ dom 𝐵 → 𝐴 ∈ V)
2		snprc 4685	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		reseq2 5971	. . . . 5 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4		res0 5980	. . . . 5 ⊢ (𝐵 ↾ ∅) = ∅
5	3, 4	eqtrdi 2820	. . . 4 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
6	2, 5	sylbi 220	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
7	6	necon1ai 2991	. 2 ⊢ ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2857	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵 ↔ 𝐴 ∈ dom 𝐵))
9		sneq 4601	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	reseq2d 5976	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
11	10	neeq1d 3023	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12		dfclel 2845	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
13	12	exbii 1875	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
14		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
15	14	eldm2 5889	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
16		n0 4314	. . . . 5 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17		elres 6017	. . . . . . 7 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18		eleq1 2857	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
19	18	pm5.32i 584	. . . . . . . . . 10 ⊢ ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20		opeq1 4839	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
21	20	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
22	21	anbi1d 642	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
23	19, 22	bitr3id 288	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
24	23	exbidv 1948	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
25	14, 24	rexsn 4650	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
26	17, 25	bitri 278	. . . . . 6 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
27	26	exbii 1875	. . . . 5 ⊢ (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
28		excom 2203	. . . . 5 ⊢ (∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
29	16, 27, 28	3bitri 300	. . . 4 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
30	13, 15, 29	3bitr4i 306	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
31	8, 11, 30	vtoclbg 3533	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
32	1, 7, 31	pm5.21nii 381	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463  ∅c0 4294  {csn 4591  ⟨cop 4597  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 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-dm 5669  df-res 5671

This theorem is referenced by:  elrn3  36149