Theorem dfhe3 41272

Description: The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
dfhe3	⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dfhe3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-he 41270	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		19.21v 1943	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
3	2	bicomi 223	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
4	3	albii 1823	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
5		alcom 2158	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
6		impexp 450	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7	6	bicomi 223	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
8	7	albii 1823	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
9		19.23v 1946	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
10	8, 9	bitri 274	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
11	10	albii 1823	. . . 4 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
12	4, 5, 11	3bitri 296	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
13		dfss2 3903	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴))
14		vex 3426	. . . . . . . 8 ⊢ 𝑦 ∈ V
15		opeq2 4802	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
16	15	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17		df-br 5071	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
18	16, 17	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑦))
19	18	anbi2d 628	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
20	19	exbidv 1925	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
21	14, 20	elab 3602	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
22	21	imbi1i 349	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
23	22	albii 1823	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
24	13, 23	bitr2i 275	. . . 4 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25		dfima3 5961	. . . . . 6 ⊢ (𝑅 “ 𝐴) = {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
26	25	eqcomi 2747	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅 “ 𝐴)
27	26	sseq1i 3945	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
28	24, 27	bitri 274	. . 3 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
29	12, 28	bitr2i 275	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
30	1, 29	bitri 274	1 ⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783   ∈ wcel 2108  {cab 2715   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   “ cima 5583   hereditary whe 41269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-he 41270

This theorem is referenced by:  psshepw  41285  dffrege69  41429