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Theorem dffr5 31942
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 3717 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2 selpw 4301 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velsn 4329 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43necon3bbii 2971 . . . . . 6 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
52, 4anbi12i 735 . . . . 5 ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
61, 5bitri 264 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
7 brdif 4849 . . . . . . 7 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥))
8 epel 5174 . . . . . . . 8 (𝑦 E 𝑥𝑦𝑥)
9 vex 3335 . . . . . . . . . . 11 𝑦 ∈ V
10 vex 3335 . . . . . . . . . . 11 𝑥 ∈ V
119, 10coep 31940 . . . . . . . . . 10 (𝑦( E ∘ 𝑅)𝑥 ↔ ∃𝑧𝑥 𝑦𝑅𝑧)
12 vex 3335 . . . . . . . . . . . 12 𝑧 ∈ V
139, 12brcnv 5452 . . . . . . . . . . 11 (𝑦𝑅𝑧𝑧𝑅𝑦)
1413rexbii 3171 . . . . . . . . . 10 (∃𝑧𝑥 𝑦𝑅𝑧 ↔ ∃𝑧𝑥 𝑧𝑅𝑦)
15 dfrex2 3126 . . . . . . . . . 10 (∃𝑧𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
1611, 14, 153bitrri 287 . . . . . . . . 9 (¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦𝑦( E ∘ 𝑅)𝑥)
1716con1bii 345 . . . . . . . 8 𝑦( E ∘ 𝑅)𝑥 ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
188, 17anbi12i 735 . . . . . . 7 ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥) ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
197, 18bitri 264 . . . . . 6 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2019exbii 1915 . . . . 5 (∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2110elrn 5513 . . . . 5 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥)
22 df-rex 3048 . . . . 5 (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2320, 21, 223bitr4i 292 . . . 4 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
246, 23imbi12i 339 . . 3 ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2524albii 1888 . 2 (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
26 dfss2 3724 . 2 ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))))
27 df-fr 5217 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2825, 26, 273bitr4ri 293 1 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1622  wex 1845  wcel 2131  wne 2924  wral 3042  wrex 3043  cdif 3704  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   class class class wbr 4796   E cep 5170   Fr wfr 5214  ccnv 5257  ran crn 5259  ccom 5262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-eprel 5171  df-fr 5217  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269
This theorem is referenced by: (None)
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