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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 4929 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵))) | |
2 | ididg 4795 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | df-br 4019 | . . . 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 2160 ⟨cop 3610 class class class wbr 4018 I cid 4303 ↾ cres 4643 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-res 4653 |
This theorem is referenced by: issref 5026 |
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