Theorem dffr5 36283

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 3916	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2		velpw 4569	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3		velsn 4607	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	necon3bbii 3007	. . . . . 6 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
5	2, 4	anbi12i 640	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
6	1, 5	bitri 278	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
7		brdif 5166	. . . . . . 7 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥))
8		epel 5566	. . . . . . . 8 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
9		vex 3461	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10		vex 3461	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	9, 10	coep 36281	. . . . . . . . . 10 ⊢ (𝑦( E ∘ ◡𝑅)𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧)
12		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	9, 12	brcnv 5870	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
14	13	rexbii 3114	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑧𝑅𝑦)
15		dfrex2 3094	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
16	11, 14, 15	3bitrri 301	. . . . . . . . 9 ⊢ (¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ 𝑦( E ∘ ◡𝑅)𝑥)
17	16	con1bii 359	. . . . . . . 8 ⊢ (¬ 𝑦( E ∘ ◡𝑅)𝑥 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
18	8, 17	anbi12i 640	. . . . . . 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 1881	. . . . 5 ⊢ (∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
21	10	elrn 5885	. . . . 5 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥)
22		df-rex 3092	. . . . 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 353	. . 3 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
25	24	albii 1852	. 2 ⊢ (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
26		df-ss 3923	. 2 ⊢ ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))))
27		df-fr 5616	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591   class class class wbr 5111   E cep 5562   Fr wfr 5613  ^◡ccnv 5662  ran crn 5664   ∘ ccom 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-eprel 5563  df-fr 5616  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674

This theorem is used by: (None)