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Mirrors > Home > MPE Home > Th. List > opabss | Structured version Visualization version 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 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} | |
2 | df-br 4584 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | eleq1 2676 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
4 | 3 | biimpar 501 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 𝑧 ∈ 𝑅) |
5 | 2, 4 | sylan2b 491 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
6 | 5 | exlimivv 1847 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
7 | 6 | abssi 3640 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} ⊆ 𝑅 |
8 | 1, 7 | eqsstri 3598 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 |
This theorem is referenced by: aceq3lem 8826 fullfunc 16389 fthfunc 16390 isfull 16393 isfth 16397 |
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