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Theorem dffr5 34330
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 3921 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2 velpw 4566 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velsn 4603 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43necon3bbii 2992 . . . . . 6 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
52, 4anbi12i 628 . . . . 5 ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
61, 5bitri 275 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
7 brdif 5159 . . . . . . 7 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥))
8 epel 5541 . . . . . . . 8 (𝑦 E 𝑥𝑦𝑥)
9 vex 3450 . . . . . . . . . . 11 𝑦 ∈ V
10 vex 3450 . . . . . . . . . . 11 𝑥 ∈ V
119, 10coep 34328 . . . . . . . . . 10 (𝑦( E ∘ 𝑅)𝑥 ↔ ∃𝑧𝑥 𝑦𝑅𝑧)
12 vex 3450 . . . . . . . . . . . 12 𝑧 ∈ V
139, 12brcnv 5839 . . . . . . . . . . 11 (𝑦𝑅𝑧𝑧𝑅𝑦)
1413rexbii 3098 . . . . . . . . . 10 (∃𝑧𝑥 𝑦𝑅𝑧 ↔ ∃𝑧𝑥 𝑧𝑅𝑦)
15 dfrex2 3077 . . . . . . . . . 10 (∃𝑧𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
1611, 14, 153bitrri 298 . . . . . . . . 9 (¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦𝑦( E ∘ 𝑅)𝑥)
1716con1bii 357 . . . . . . . 8 𝑦( E ∘ 𝑅)𝑥 ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
188, 17anbi12i 628 . . . . . . 7 ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥) ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
197, 18bitri 275 . . . . . 6 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2019exbii 1851 . . . . 5 (∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2110elrn 5850 . . . . 5 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥)
22 df-rex 3075 . . . . 5 (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2320, 21, 223bitr4i 303 . . . 4 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
246, 23imbi12i 351 . . 3 ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2524albii 1822 . 2 (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
26 dfss2 3931 . 2 ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))))
27 df-fr 5589 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2825, 26, 273bitr4ri 304 1 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  wne 2944  wral 3065  wrex 3074  cdif 3908  wss 3911  c0 4283  𝒫 cpw 4561  {csn 4587   class class class wbr 5106   E cep 5537   Fr wfr 5586  ccnv 5633  ran crn 5635  ccom 5638
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-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-fr 5589  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645
This theorem is referenced by: (None)
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