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Theorem dfhe3 42511
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 42509 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 19.21v 1942 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
32bicomi 223 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
43albii 1821 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
5 alcom 2156 . . . 4 (∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
6 impexp 451 . . . . . . . 8 (((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
76bicomi 223 . . . . . . 7 ((𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
87albii 1821 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
9 19.23v 1945 . . . . . 6 (∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
108, 9bitri 274 . . . . 5 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
1110albii 1821 . . . 4 (∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
124, 5, 113bitri 296 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
13 dfss2 3967 . . . . 5 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴))
14 vex 3478 . . . . . . . 8 𝑦 ∈ V
15 opeq2 4873 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2818 . . . . . . . . . . 11 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17 df-br 5148 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17bitr4di 288 . . . . . . . . . 10 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑦))
1918anbi2d 629 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥𝐴𝑥𝑅𝑦)))
2019exbidv 1924 . . . . . . . 8 (𝑧 = 𝑦 → (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦)))
2114, 20elab 3667 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
2221imbi1i 349 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2322albii 1821 . . . . 5 (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2413, 23bitr2i 275 . . . 4 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25 dfima3 6060 . . . . . 6 (𝑅𝐴) = {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
2625eqcomi 2741 . . . . 5 {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅𝐴)
2726sseq1i 4009 . . . 4 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2824, 27bitri 274 . . 3 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑅𝐴) ⊆ 𝐴)
2912, 28bitr2i 275 . 2 ((𝑅𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
301, 29bitri 274 1 (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781  wcel 2106  {cab 2709  wss 3947  cop 4633   class class class wbr 5147  cima 5678   hereditary whe 42508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-he 42509
This theorem is referenced by:  psshepw  42524  dffrege69  42668
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