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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 4637 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ (〈𝐴, 𝐴〉 ∈ I ∧ 𝐴 ∈ 𝐵))) | |
2 | ididg 4507 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | df-br 3786 | . . . 4 ⊢ (𝐴 I 𝐴 ↔ 〈𝐴, 𝐴〉 ∈ I ) | |
4 | 2, 3 | sylib 120 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 ∈ I ) |
5 | 4 | biantrurd 299 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (〈𝐴, 𝐴〉 ∈ I ∧ 𝐴 ∈ 𝐵))) |
6 | 1, 5 | bitr4d 189 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 I cid 4043 ↾ cres 4365 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-res 4375 |
This theorem is referenced by: issref 4727 |
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