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Theorem dfhe3 38386
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(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dfhe3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-he 38384 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 19.21v 1908 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
32bicomi 214 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
43albii 1787 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
5 alcom 2077 . . . 4 (∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
6 impexp 461 . . . . . . . 8 (((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
76bicomi 214 . . . . . . 7 ((𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
87albii 1787 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
9 19.23v 1911 . . . . . 6 (∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
108, 9bitri 264 . . . . 5 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
1110albii 1787 . . . 4 (∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
124, 5, 113bitri 286 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
13 dfss2 3624 . . . . 5 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴))
14 vex 3234 . . . . . . . 8 𝑦 ∈ V
15 opeq2 4434 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2715 . . . . . . . . . . 11 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17 df-br 4686 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17syl6bbr 278 . . . . . . . . . 10 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑦))
1918anbi2d 740 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥𝐴𝑥𝑅𝑦)))
2019exbidv 1890 . . . . . . . 8 (𝑧 = 𝑦 → (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦)))
2114, 20elab 3382 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
2221imbi1i 338 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2322albii 1787 . . . . 5 (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2413, 23bitr2i 265 . . . 4 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25 dfima3 5504 . . . . . 6 (𝑅𝐴) = {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
2625eqcomi 2660 . . . . 5 {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅𝐴)
2726sseq1i 3662 . . . 4 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2824, 27bitri 264 . . 3 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑅𝐴) ⊆ 𝐴)
2912, 28bitr2i 265 . 2 ((𝑅𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
301, 29bitri 264 1 (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744  wcel 2030  {cab 2637  wss 3607  cop 4216   class class class wbr 4685  cima 5146   hereditary whe 38383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-he 38384
This theorem is referenced by:  psshepw  38399  dffrege69  38543
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