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Theorem dffr5 30896
Description: A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
Assertion
Ref Expression
dffr5 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))

Proof of Theorem dffr5
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3550 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2 selpw 4115 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velsn 4141 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43necon3bbii 2829 . . . . . 6 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
52, 4anbi12i 729 . . . . 5 ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
61, 5bitri 263 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
7 brdif 4635 . . . . . . 7 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥))
8 epel 4952 . . . . . . . 8 (𝑦 E 𝑥𝑦𝑥)
9 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
10 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
119, 10coep 30894 . . . . . . . . . 10 (𝑦( E ∘ 𝑅)𝑥 ↔ ∃𝑧𝑥 𝑦𝑅𝑧)
12 vex 3176 . . . . . . . . . . . 12 𝑧 ∈ V
139, 12brcnv 5227 . . . . . . . . . . 11 (𝑦𝑅𝑧𝑧𝑅𝑦)
1413rexbii 3023 . . . . . . . . . 10 (∃𝑧𝑥 𝑦𝑅𝑧 ↔ ∃𝑧𝑥 𝑧𝑅𝑦)
15 dfrex2 2979 . . . . . . . . . 10 (∃𝑧𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
1611, 14, 153bitrri 286 . . . . . . . . 9 (¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦𝑦( E ∘ 𝑅)𝑥)
1716con1bii 345 . . . . . . . 8 𝑦( E ∘ 𝑅)𝑥 ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
188, 17anbi12i 729 . . . . . . 7 ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥) ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
197, 18bitri 263 . . . . . 6 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2019exbii 1764 . . . . 5 (∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2110elrn 5287 . . . . 5 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥)
22 df-rex 2902 . . . . 5 (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2320, 21, 223bitr4i 291 . . . 4 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
246, 23imbi12i 339 . . 3 ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2524albii 1737 . 2 (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
26 dfss2 3557 . 2 ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))))
27 df-fr 4997 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2825, 26, 273bitr4ri 292 1 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  wal 1473  wex 1695  wcel 1977  wne 2780  wral 2896  wrex 2897  cdif 3537  wss 3540  c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   E cep 4947   Fr wfr 4994  ccnv 5037  ran crn 5039  ccom 5042
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-eu 2462  df-mo 2463  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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049
This theorem is referenced by: (None)
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