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Theorem dffr5 32873
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 3950 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2 velpw 4550 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velsn 4580 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43necon3bbii 3068 . . . . . 6 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
52, 4anbi12i 626 . . . . 5 ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
61, 5bitri 276 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
7 brdif 5116 . . . . . . 7 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥))
8 epel 5468 . . . . . . . 8 (𝑦 E 𝑥𝑦𝑥)
9 vex 3503 . . . . . . . . . . 11 𝑦 ∈ V
10 vex 3503 . . . . . . . . . . 11 𝑥 ∈ V
119, 10coep 32871 . . . . . . . . . 10 (𝑦( E ∘ 𝑅)𝑥 ↔ ∃𝑧𝑥 𝑦𝑅𝑧)
12 vex 3503 . . . . . . . . . . . 12 𝑧 ∈ V
139, 12brcnv 5752 . . . . . . . . . . 11 (𝑦𝑅𝑧𝑧𝑅𝑦)
1413rexbii 3252 . . . . . . . . . 10 (∃𝑧𝑥 𝑦𝑅𝑧 ↔ ∃𝑧𝑥 𝑧𝑅𝑦)
15 dfrex2 3244 . . . . . . . . . 10 (∃𝑧𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
1611, 14, 153bitrri 299 . . . . . . . . 9 (¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦𝑦( E ∘ 𝑅)𝑥)
1716con1bii 358 . . . . . . . 8 𝑦( E ∘ 𝑅)𝑥 ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
188, 17anbi12i 626 . . . . . . 7 ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥) ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
197, 18bitri 276 . . . . . 6 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2019exbii 1841 . . . . 5 (∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2110elrn 5821 . . . . 5 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥)
22 df-rex 3149 . . . . 5 (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2320, 21, 223bitr4i 304 . . . 4 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
246, 23imbi12i 352 . . 3 ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2524albii 1813 . 2 (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
26 dfss2 3959 . 2 ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))))
27 df-fr 5513 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2825, 26, 273bitr4ri 305 1 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  cdif 3937  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   class class class wbr 5063   E cep 5463   Fr wfr 5510  ccnv 5553  ran crn 5555  ccom 5558
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 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-eprel 5464  df-fr 5513  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565
This theorem is referenced by: (None)
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