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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 5212 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} | |
2 | df-br 5150 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | eleq1 2813 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
4 | 3 | biimpar 476 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 𝑧 ∈ 𝑅) |
5 | 2, 4 | sylan2b 592 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
6 | 5 | exlimivv 1927 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝑧 ∈ 𝑅) |
7 | 6 | abssi 4063 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)} ⊆ 𝑅 |
8 | 1, 7 | eqsstri 4011 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 ⊆ wss 3944 〈cop 4636 class class class wbr 5149 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ss 3961 df-br 5150 df-opab 5212 |
This theorem is referenced by: elopabr 5563 elopabran 5564 mptmpoopabbrd 8085 aceq3lem 10145 fullfunc 17898 fthfunc 17899 isfull 17902 isfth 17906 wksv 29505 |
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