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Theorem eldm3 35741
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.
StepHypRef Expression
1 elex 3499 . 2 (𝐴 ∈ dom 𝐵𝐴 ∈ V)
2 snprc 4722 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 reseq2 5995 . . . . 5 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4 res0 6004 . . . . 5 (𝐵 ↾ ∅) = ∅
53, 4eqtrdi 2791 . . . 4 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
62, 5sylbi 217 . . 3 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
76necon1ai 2966 . 2 ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2827 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵𝐴 ∈ dom 𝐵))
9 sneq 4641 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
109reseq2d 6000 . . . 4 (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
1110neeq1d 2998 . . 3 (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12 dfclel 2815 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
1312exbii 1845 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
14 vex 3482 . . . . 5 𝑥 ∈ V
1514eldm2 5915 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
16 n0 4359 . . . . 5 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17 elres 6040 . . . . . . 7 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18 eleq1 2827 . . . . . . . . . . 11 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
1918pm5.32i 574 . . . . . . . . . 10 ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20 opeq1 4878 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2120eqeq2d 2746 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
2221anbi1d 631 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2319, 22bitr3id 285 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2423exbidv 1919 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2514, 24rexsn 4687 . . . . . . 7 (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2617, 25bitri 275 . . . . . 6 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2726exbii 1845 . . . . 5 (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
28 excom 2160 . . . . 5 (∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵) ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2916, 27, 283bitri 297 . . . 4 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
3013, 15, 293bitr4i 303 . . 3 (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
318, 11, 30vtoclbg 3557 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
321, 7, 31pm5.21nii 378 1 (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  c0 4339  {csn 4631  cop 4637  dom cdm 5689  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701
This theorem is referenced by:  elrn3  35742
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