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Mirrors > Home > MPE Home > Th. List > opres | Structured version Visualization version GIF version |
Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
opres.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opres | ⊢ (𝐴 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opres.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | opelresi 5854 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (𝐴 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
3 | 2 | baib 536 | 1 ⊢ (𝐴 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 〈cop 4563 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-res 5560 |
This theorem is referenced by: resieq 5857 2elresin 6461 mdetunilem9 21157 |
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