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Theorem dfhe3 40256
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 40254 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2 19.21v 1940 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
32bicomi 226 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
43albii 1820 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
5 alcom 2163 . . . 4 (∀𝑥𝑦(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
6 impexp 453 . . . . . . . 8 (((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)))
76bicomi 226 . . . . . . 7 ((𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
87albii 1820 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
9 19.23v 1943 . . . . . 6 (∀𝑥((𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
108, 9bitri 277 . . . . 5 (∀𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
1110albii 1820 . . . 4 (∀𝑦𝑥(𝑥𝐴 → (𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
124, 5, 113bitri 299 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
13 dfss2 3933 . . . . 5 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴))
14 vex 3476 . . . . . . . 8 𝑦 ∈ V
15 opeq2 4780 . . . . . . . . . . . 12 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2895 . . . . . . . . . . 11 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17 df-br 5043 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17syl6bbr 291 . . . . . . . . . 10 (𝑧 = 𝑦 → (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑦))
1918anbi2d 630 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (𝑥𝐴𝑥𝑅𝑦)))
2019exbidv 1922 . . . . . . . 8 (𝑧 = 𝑦 → (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦)))
2114, 20elab 3647 . . . . . . 7 (𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
2221imbi1i 352 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2322albii 1820 . . . . 5 (∀𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} → 𝑦𝐴) ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴))
2413, 23bitr2i 278 . . . 4 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴)
25 dfima3 5908 . . . . . 6 (𝑅𝐴) = {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)}
2625eqcomi 2829 . . . . 5 {𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} = (𝑅𝐴)
2726sseq1i 3974 . . . 4 ({𝑧 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑅)} ⊆ 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
2824, 27bitri 277 . . 3 (∀𝑦(∃𝑥(𝑥𝐴𝑥𝑅𝑦) → 𝑦𝐴) ↔ (𝑅𝐴) ⊆ 𝐴)
2912, 28bitr2i 278 . 2 ((𝑅𝐴) ⊆ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
301, 29bitri 277 1 (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1535  wex 1780  wcel 2114  {cab 2798  wss 3913  cop 4549   class class class wbr 5042  cima 5534   hereditary whe 40253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-he 40254
This theorem is referenced by:  psshepw  40269  dffrege69  40413
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