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Mirrors > Home > ILE Home > Th. List > opabss | GIF version |
Description: The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
opabss | ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 4051 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} | |
2 | df-br 3990 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | eleq1 2233 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
4 | 3 | biimpar 295 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 𝑧 ∈ 𝑅) |
5 | 2, 4 | sylan2b 285 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
6 | 5 | exlimivv 1889 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
7 | 6 | abssi 3222 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} ⊆ 𝑅 |
8 | 1, 7 | eqsstri 3179 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 ⊆ wss 3121 〈cop 3586 class class class wbr 3989 {copab 4049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 df-br 3990 df-opab 4051 |
This theorem is referenced by: (None) |
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