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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 6008 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (𝐴 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
3 | 2 | baib 535 | 1 ⊢ (𝐴 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-res 5701 |
This theorem is referenced by: resieq 6011 2elresin 6690 mdetunilem9 22642 |
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