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Theorem eldm3 31777
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 3243 . 2 (𝐴 ∈ dom 𝐵𝐴 ∈ V)
2 snprc 4285 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 reseq2 5423 . . . . 5 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4 res0 5432 . . . . 5 (𝐵 ↾ ∅) = ∅
53, 4syl6eq 2701 . . . 4 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
62, 5sylbi 207 . . 3 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
76necon1ai 2850 . 2 ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2718 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵𝐴 ∈ dom 𝐵))
9 sneq 4220 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
109reseq2d 5428 . . . 4 (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
1110neeq1d 2882 . . 3 (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12 df-clel 2647 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
1312exbii 1814 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
14 vex 3234 . . . . 5 𝑥 ∈ V
1514eldm2 5354 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
16 n0 3964 . . . . 5 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17 elres 5470 . . . . . . 7 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18 eleq1 2718 . . . . . . . . . . 11 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
1918pm5.32i 670 . . . . . . . . . 10 ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20 opeq1 4433 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2120eqeq2d 2661 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
2221anbi1d 741 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2319, 22syl5bbr 274 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2423exbidv 1890 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2514, 24rexsn 4255 . . . . . . 7 (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2617, 25bitri 264 . . . . . 6 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2726exbii 1814 . . . . 5 (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
28 excom 2082 . . . . 5 (∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵) ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2916, 27, 283bitri 286 . . . 4 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
3013, 15, 293bitr4i 292 . . 3 (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
318, 11, 30vtoclbg 3298 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
321, 7, 31pm5.21nii 367 1 (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  c0 3948  {csn 4210  cop 4216  dom cdm 5143  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-res 5155
This theorem is referenced by:  elrn3  31778
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