Theorem dfhe3 37089

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 37087	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		19.21v 1855	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
3	2	bicomi 213	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
4	3	albii 1737	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
5		alcom 2024	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
6		impexp 461	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7	6	bicomi 213	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
8	7	albii 1737	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
9		19.23v 1889	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
10	8, 9	bitri 263	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
11	10	albii 1737	. . . 4 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
12	4, 5, 11	3bitri 285	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
13		dfss2 3557	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴))
14		vex 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
15		opeq2 4341	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
16	15	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17		df-br 4584	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
18	16, 17	syl6bbr 277	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑦))
19	18	anbi2d 736	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
20	19	exbidv 1837	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
21	14, 20	elab 3319	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
22	21	imbi1i 338	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
23	22	albii 1737	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ ∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
24	13, 23	bitr2i 264	. . . 4 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25		dfima3 5388	. . . . . 6 ⊢ (𝑅 “ 𝐴) = {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
26	25	eqcomi 2619	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅 “ 𝐴)
27	26	sseq1i 3592	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
28	24, 27	bitri 263	. . 3 ⊢ (∀𝑦(∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴) ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
29	12, 28	bitr2i 264	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
30	1, 29	bitri 263	1 ⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  {cab 2596   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   “ cima 5041   hereditary whe 37086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087

This theorem is referenced by:  psshepw  37102  dffrege69  37246