Theorem dffr5 36333

Description: A quantifier-free definition of a well-founded relation. (Contributed by Scott Fenton, 11-Apr-2011.) (Proof shortened by Scott Fenton, 26-Aug-2026.)

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		brdif 5158	. . . . . 6 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥))
2		epel 5558	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
3		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
5	3, 4	coep 36331	. . . . . . . . 9 ⊢ (𝑦( E ∘ ◡𝑅)𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧)
6		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	3, 6	brcnv 5862	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
8	7	rexbii 3109	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑥 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑧𝑅𝑦)
9		dfrex2 3089	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑥 𝑧𝑅𝑦 ↔ ¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
10	5, 8, 9	3bitrri 301	. . . . . . . 8 ⊢ (¬ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ 𝑦( E ∘ ◡𝑅)𝑥)
11	10	con1bii 359	. . . . . . 7 ⊢ (¬ 𝑦( E ∘ ◡𝑅)𝑥 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
12	2, 11	anbi12i 640	. . . . . 6 ⊢ ((𝑦 E 𝑥 ∧ ¬ 𝑦( E ∘ ◡𝑅)𝑥) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
13	1, 12	bitri 278	. . . . 5 ⊢ (𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
14	13	exbii 1881	. . . 4 ⊢ (∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
15	4	elrn 5877	. . . 4 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 𝑦( E ∖ ( E ∘ ◡𝑅))𝑥)
16		df-rex 3087	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
17	14, 15, 16	3bitr4i 306	. . 3 ⊢ (𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
18	17	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
19		dfss3 3920	. 2 ⊢ ((𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)) ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})𝑥 ∈ ran ( E ∖ ( E ∘ ◡𝑅)))
20		dffr6 5611	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
21	18, 19, 20	3bitr4ri 307	1 ⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   E cep 5554   Fr wfr 5605  ^◡ccnv 5654  ran crn 5656   ∘ ccom 5659

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5555  df-fr 5608  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666

This theorem is used by: (None)