Theorem dfhe3 40119

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 40117	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		19.21v 1936	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
3	2	bicomi 226	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
4	3	albii 1816	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
5		alcom 2159	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
6		impexp 453	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7	6	bicomi 226	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
8	7	albii 1816	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
9		19.23v 1939	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
10	8, 9	bitri 277	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
11	10	albii 1816	. . . 4 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
12	4, 5, 11	3bitri 299	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
13		dfss2 3954	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴))
14		vex 3497	. . . . . . . 8 ⊢ 𝑦 ∈ V
15		opeq2 4803	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
16	15	eleq1d 2897	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17		df-br 5066	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
18	16, 17	syl6bbr 291	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑦))
19	18	anbi2d 630	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
20	19	exbidv 1918	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
21	14, 20	elab 3666	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
22	21	imbi1i 352	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
23	22	albii 1816	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
24	13, 23	bitr2i 278	. . . 4 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25		dfima3 5931	. . . . . 6 ⊢ (𝑅 “ 𝐴) = {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
26	25	eqcomi 2830	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅 “ 𝐴)
27	26	sseq1i 3994	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
28	24, 27	bitri 277	. . 3 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
29	12, 28	bitr2i 278	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
30	1, 29	bitri 277	1 ⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776   ∈ wcel 2110  {cab 2799   ⊆ wss 3935  ⟨cop 4572   class class class wbr 5065   “ cima 5557   hereditary whe 40116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-he 40117

This theorem is referenced by:  psshepw  40132  dffrege69  40276