Theorem eldm3 30905

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 3185	. 2 ⊢ (𝐴 ∈ dom 𝐵 → 𝐴 ∈ V)
2		snprc 4197	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		reseq2 5312	. . . . 5 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4		res0 5321	. . . . 5 ⊢ (𝐵 ↾ ∅) = ∅
5	3, 4	syl6eq 2660	. . . 4 ⊢ ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
6	2, 5	sylbi 206	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
7	6	necon1ai 2809	. 2 ⊢ ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2676	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵 ↔ 𝐴 ∈ dom 𝐵))
9		sneq 4135	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	reseq2d 5317	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
11	10	neeq1d 2841	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12		df-clel 2606	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
13	12	exbii 1764	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
14		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
15	14	eldm2 5244	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
16		n0 3890	. . . . 5 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17		elres 5355	. . . . . . 7 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
19	18	pm5.32i 667	. . . . . . . . . 10 ⊢ ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20		opeq1 4340	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
21	20	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
22	21	anbi1d 737	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
23	19, 22	syl5bbr 273	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
24	23	exbidv 1837	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵)))
25	14, 24	rexsn 4170	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
26	17, 25	bitri 263	. . . . . 6 ⊢ (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
27	26	exbii 1764	. . . . 5 ⊢ (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
28		excom 2029	. . . . 5 ⊢ (∃𝑝∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵) ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
29	16, 27, 28	3bitri 285	. . . 4 ⊢ ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝 ∈ 𝐵))
30	13, 15, 29	3bitr4i 291	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
31	8, 11, 30	vtoclbg 3240	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
32	1, 7, 31	pm5.21nii 367	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131  dom cdm 5038   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050

This theorem is referenced by:  elrn3  30906