Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldm3 Structured version   Visualization version   GIF version

Theorem eldm3 34731
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 3493 . 2 (𝐴 ∈ dom 𝐵𝐴 ∈ V)
2 snprc 4722 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 reseq2 5977 . . . . 5 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4 res0 5986 . . . . 5 (𝐵 ↾ ∅) = ∅
53, 4eqtrdi 2789 . . . 4 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
62, 5sylbi 216 . . 3 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
76necon1ai 2969 . 2 ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2822 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵𝐴 ∈ dom 𝐵))
9 sneq 4639 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
109reseq2d 5982 . . . 4 (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
1110neeq1d 3001 . . 3 (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12 dfclel 2812 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
1312exbii 1851 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
14 vex 3479 . . . . 5 𝑥 ∈ V
1514eldm2 5902 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
16 n0 4347 . . . . 5 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17 elres 6021 . . . . . . 7 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18 eleq1 2822 . . . . . . . . . . 11 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
1918pm5.32i 576 . . . . . . . . . 10 ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20 opeq1 4874 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2120eqeq2d 2744 . . . . . . . . . . 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 1925 . . . . . . . 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 1851 . . . . 5 (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
28 excom 2163 . . . . 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 3560 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
321, 7, 31pm5.21nii 380 1 (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  Vcvv 3475  c0 4323  {csn 4629  cop 4635  dom cdm 5677  cres 5679
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-res 5689
This theorem is referenced by:  elrn3  34732
  Copyright terms: Public domain W3C validator