Theorem dfhe3 44312

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 44310	. 2 ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
2		19.21v 1958	. . . . . 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 1838	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
5		alcom 2192	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
6		impexp 454	. . . . . . . 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 1838	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
9		19.23v 1961	. . . . . 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 1838	. . . 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		df-ss 3919	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴))
14		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
15		opeq2 4829	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
16	15	eleq1d 2846	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17		df-br 5098	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
18	16, 17	bitr4di 291	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑦))
19	18	anbi2d 639	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
20	19	exbidv 1940	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
21	14, 20	elab 3637	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
22	21	imbi1i 351	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦 ∈ 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝐴))
23	22	albii 1838	. . . . 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 6048	. . . . . 6 ⊢ (𝑅 “ 𝐴) = {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
26	25	eqcomi 2770	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅 “ 𝐴)
27	26	sseq1i 3962	. . . 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 399  ∀wal 1557  ∃wex 1798   ∈ wcel 2141  {cab 2739   ⊆ wss 3902  ⟨cop 4585   class class class wbr 5097   “ cima 5646   hereditary whe 44309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-he 44310

This theorem is referenced by:  psshepw  44325  dffrege69  44469