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Mirrors > Home > ILE Home > Th. List > opelresi | GIF version |
Description: ⟨𝐴, 𝐴⟩ belongs to a restriction of the identity class iff 𝐴 belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.) |
Ref | Expression |
---|---|
opelresi | ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelresg 4926 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵))) | |
2 | ididg 4792 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | df-br 4016 | . . . 4 ⊢ (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I ) | |
4 | 2, 3 | sylib 122 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ ∈ I ) |
5 | 4 | biantrurd 305 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵))) |
6 | 1, 5 | bitr4d 191 | 1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2158 ⟨cop 3607 class class class wbr 4015 I cid 4300 ↾ cres 4640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-res 4650 |
This theorem is referenced by: issref 5023 |
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