Theorem dffr5 33102

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.

Step	Hyp	Ref	Expression
1		eldif 3891	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2		velpw 4502	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3		velsn 4541	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	necon3bbii 3034	. . . . . 6 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
5	2, 4	anbi12i 629	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
6	1, 5	bitri 278	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
7		brdif 5083	. . . . . . 7 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥))
8		epel 5433	. . . . . . . 8 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
9		vex 3444	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10		vex 3444	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	9, 10	coep 33100	. . . . . . . . . 10 ⊢ (𝑦( E ∘ ◡𝑅)𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧)
12		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	9, 12	brcnv 5717	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
14	13	rexbii 3210	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑧𝑅𝑦)
15		dfrex2 3202	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
16	11, 14, 15	3bitrri 301	. . . . . . . . 9 ⊢ (¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ 𝑦( E ∘ ◡𝑅)𝑥)
17	16	con1bii 360	. . . . . . . 8 ⊢ (¬ 𝑦( E ∘ ◡𝑅)𝑥 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
18	8, 17	anbi12i 629	. . . . . . 7 ⊢ ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
19	7, 18	bitri 278	. . . . . 6 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
20	19	exbii 1849	. . . . 5 ⊢ (∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
21	10	elrn 5786	. . . . 5 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥)
22		df-rex 3112	. . . . 5 ⊢ (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
23	20, 21, 22	3bitr4i 306	. . . 4 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
24	6, 23	imbi12i 354	. . 3 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
25	24	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
26		dfss2 3901	. 2 ⊢ ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))))
27		df-fr 5478	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
28	25, 26, 27	3bitr4ri 307	1 ⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525   class class class wbr 5030   E cep 5429   Fr wfr 5475  ^◡ccnv 5518  ran crn 5520   ∘ ccom 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-fr 5478  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530

This theorem is referenced by: (None)