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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 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} | |
| 2 | df-br 5144 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
| 3 | eleq1 2829 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
| 4 | 3 | biimpar 477 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 𝑧 ∈ 𝑅) |
| 5 | 2, 4 | sylan2b 594 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
| 6 | 5 | exlimivv 1932 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
| 7 | 6 | abssi 4070 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} ⊆ 𝑅 |
| 8 | 1, 7 | eqsstri 4030 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 |
| This theorem is referenced by: elopabr 5566 elopabran 5567 mptmpoopabbrd 8105 aceq3lem 10160 fullfunc 17953 fthfunc 17954 isfull 17957 isfth 17961 wksv 29637 |
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