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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 4733 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ (〈𝐴, 𝐴〉 ∈ I ∧ 𝐴 ∈ 𝐵))) | |
2 | ididg 4602 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | df-br 3852 | . . . 4 ⊢ (𝐴 I 𝐴 ↔ 〈𝐴, 𝐴〉 ∈ I ) | |
4 | 2, 3 | sylib 121 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 ∈ I ) |
5 | 4 | biantrurd 300 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (〈𝐴, 𝐴〉 ∈ I ∧ 𝐴 ∈ 𝐵))) |
6 | 1, 5 | bitr4d 190 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1439 〈cop 3453 class class class wbr 3851 I cid 4124 ↾ cres 4454 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-res 4464 |
This theorem is referenced by: issref 4827 |
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