Theorem dffr5 36101

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 3914	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
2		velpw 4560	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3		velsn 4598	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	necon3bbii 3004	. . . . . 6 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ≠ ∅)
5	2, 4	anbi12i 637	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
6	1, 5	bitri 277	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
7		brdif 5153	. . . . . . 7 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥))
8		epel 5550	. . . . . . . 8 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
9		vex 3458	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	9, 10	coep 36099	. . . . . . . . . 10 ⊢ (𝑦( E ∘ ◡𝑅)𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧)
12		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	9, 12	brcnv 5854	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
14	13	rexbii 3109	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑧𝑅𝑦)
15		dfrex2 3089	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
16	11, 14, 15	3bitrri 300	. . . . . . . . 9 ⊢ (¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ 𝑦( E ∘ ◡𝑅)𝑥)
17	16	con1bii 358	. . . . . . . 8 ⊢ (¬ 𝑦( E ∘ ◡𝑅)𝑥 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
18	8, 17	anbi12i 637	. . . . . . 7 ⊢ ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
19	7, 18	bitri 277	. . . . . 6 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
20	19	exbii 1868	. . . . 5 ⊢ (∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
21	10	elrn 5869	. . . . 5 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥)
22		df-rex 3087	. . . . 5 ⊢ (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
23	20, 21, 22	3bitr4i 305	. . . 4 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
24	6, 23	imbi12i 352	. . 3 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
25	24	albii 1839	. 2 ⊢ (∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
26		df-ss 3921	. 2 ⊢ ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅))))
27		df-fr 5600	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
28	25, 26, 27	3bitr4ri 306	1 ⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  {csn 4582   class class class wbr 5100   E cep 5546   Fr wfr 5597  ^◡ccnv 5646  ran crn 5648   ∘ ccom 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5547  df-fr 5600  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658

This theorem is referenced by: (None)