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Theorem dffr5 32088
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 3742 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2 selpw 4322 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velsn 4350 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43necon3bbii 2984 . . . . . 6 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
52, 4anbi12i 620 . . . . 5 ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
61, 5bitri 266 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
7 brdif 4862 . . . . . . 7 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥))
8 epel 5193 . . . . . . . 8 (𝑦 E 𝑥𝑦𝑥)
9 vex 3353 . . . . . . . . . . 11 𝑦 ∈ V
10 vex 3353 . . . . . . . . . . 11 𝑥 ∈ V
119, 10coep 32086 . . . . . . . . . 10 (𝑦( E ∘ 𝑅)𝑥 ↔ ∃𝑧𝑥 𝑦𝑅𝑧)
12 vex 3353 . . . . . . . . . . . 12 𝑧 ∈ V
139, 12brcnv 5473 . . . . . . . . . . 11 (𝑦𝑅𝑧𝑧𝑅𝑦)
1413rexbii 3188 . . . . . . . . . 10 (∃𝑧𝑥 𝑦𝑅𝑧 ↔ ∃𝑧𝑥 𝑧𝑅𝑦)
15 dfrex2 3142 . . . . . . . . . 10 (∃𝑧𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
1611, 14, 153bitrri 289 . . . . . . . . 9 (¬ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦𝑦( E ∘ 𝑅)𝑥)
1716con1bii 347 . . . . . . . 8 𝑦( E ∘ 𝑅)𝑥 ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
188, 17anbi12i 620 . . . . . . 7 ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ 𝑅)𝑥) ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
197, 18bitri 266 . . . . . 6 (𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ (𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2019exbii 1943 . . . . 5 (∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2110elrn 5535 . . . . 5 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ 𝑅))𝑥)
22 df-rex 3061 . . . . 5 (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
2320, 21, 223bitr4i 294 . . . 4 (𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
246, 23imbi12i 341 . . 3 ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2524albii 1914 . 2 (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
26 dfss2 3749 . 2 ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ 𝑅))))
27 df-fr 5236 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2825, 26, 273bitr4ri 295 1 (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  cdif 3729  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   class class class wbr 4809   E cep 5189   Fr wfr 5233  ccnv 5276  ran crn 5278  ccom 5281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-eprel 5190  df-fr 5236  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288
This theorem is referenced by: (None)
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