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Theorem dfhe3 37586
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 37584 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 19.21v 1865 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
32bicomi 214 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
43albii 1744 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
5 alcom 2034 . . . 4 (∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
6 impexp 462 . . . . . . . 8 (((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
76bicomi 214 . . . . . . 7 ((𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
87albii 1744 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
9 19.23v 1899 . . . . . 6 (∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
108, 9bitri 264 . . . . 5 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
1110albii 1744 . . . 4 (∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
124, 5, 113bitri 286 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
13 dfss2 3576 . . . . 5 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴))
14 vex 3192 . . . . . . . 8 𝑦 ∈ V
15 opeq2 4376 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2683 . . . . . . . . . . 11 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17 df-br 4619 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17syl6bbr 278 . . . . . . . . . 10 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑦))
1918anbi2d 739 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥𝐴𝑥𝑅𝑦)))
2019exbidv 1847 . . . . . . . 8 (𝑧 = 𝑦 → (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦)))
2114, 20elab 3337 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
2221imbi1i 339 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2322albii 1744 . . . . 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 5433 . . . . . 6 (𝑅𝐴) = {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
2625eqcomi 2630 . . . . 5 {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅𝐴)
2726sseq1i 3613 . . . 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 384  wal 1478  wex 1701  wcel 1987  {cab 2607  wss 3559  cop 4159   class class class wbr 4618  cima 5082   hereditary whe 37583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-he 37584
This theorem is referenced by:  psshepw  37599  dffrege69  37743
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