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Theorem eldm3 32985
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 3511 . 2 (𝐴 ∈ dom 𝐵𝐴 ∈ V)
2 snprc 4645 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 reseq2 5841 . . . . 5 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = (𝐵 ↾ ∅))
4 res0 5850 . . . . 5 (𝐵 ↾ ∅) = ∅
53, 4syl6eq 2870 . . . 4 ({𝐴} = ∅ → (𝐵 ↾ {𝐴}) = ∅)
62, 5sylbi 219 . . 3 𝐴 ∈ V → (𝐵 ↾ {𝐴}) = ∅)
76necon1ai 3041 . 2 ((𝐵 ↾ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2898 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐵𝐴 ∈ dom 𝐵))
9 sneq 4569 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
109reseq2d 5846 . . . 4 (𝑥 = 𝐴 → (𝐵 ↾ {𝑥}) = (𝐵 ↾ {𝐴}))
1110neeq1d 3073 . . 3 (𝑥 = 𝐴 → ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
12 dfclel 2892 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
1312exbii 1841 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
14 vex 3496 . . . . 5 𝑥 ∈ V
1514eldm2 5763 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
16 n0 4308 . . . . 5 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}))
17 elres 5884 . . . . . . 7 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
18 eleq1 2898 . . . . . . . . . . 11 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
1918pm5.32i 577 . . . . . . . . . 10 ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
20 opeq1 4795 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2120eqeq2d 2830 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑝 = ⟨𝑧, 𝑦⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
2221anbi1d 631 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2319, 22syl5bbr 287 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2423exbidv 1915 . . . . . . . 8 (𝑧 = 𝑥 → (∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵)))
2514, 24rexsn 4612 . . . . . . 7 (∃𝑧 ∈ {𝑥}∃𝑦(𝑝 = ⟨𝑧, 𝑦⟩ ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2617, 25bitri 277 . . . . . 6 (𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2726exbii 1841 . . . . 5 (∃𝑝 𝑝 ∈ (𝐵 ↾ {𝑥}) ↔ ∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
28 excom 2161 . . . . 5 (∃𝑝𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵) ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
2916, 27, 283bitri 299 . . . 4 ((𝐵 ↾ {𝑥}) ≠ ∅ ↔ ∃𝑦𝑝(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑝𝐵))
3013, 15, 293bitr4i 305 . . 3 (𝑥 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝑥}) ≠ ∅)
318, 11, 30vtoclbg 3567 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅))
321, 7, 31pm5.21nii 382 1 (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  wne 3014  wrex 3137  Vcvv 3493  c0 4289  {csn 4559  cop 4565  dom cdm 5548  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-res 5560
This theorem is referenced by:  elrn3  32986
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