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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 5093 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} | |
2 | df-br 5031 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | eleq1 2877 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
4 | 3 | biimpar 481 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 𝑧 ∈ 𝑅) |
5 | 2, 4 | sylan2b 596 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
6 | 5 | exlimivv 1933 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
7 | 6 | abssi 3997 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} ⊆ 𝑅 |
8 | 1, 7 | eqsstri 3949 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ⊆ wss 3881 〈cop 4531 class class class wbr 5030 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 |
This theorem is referenced by: aceq3lem 9531 fullfunc 17168 fthfunc 17169 isfull 17172 isfth 17176 |
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