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Mirrors > Home > MPE Home > Th. List > Mathboxes > brresi2 | Structured version Visualization version GIF version |
Description: Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
brresi2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brresi2 | ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 5851 | . 2 ⊢ (𝑅 ↾ 𝐶) ⊆ 𝑅 | |
2 | 1 | ssbri 5076 | 1 ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Vcvv 3398 class class class wbr 5031 ↾ cres 5528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-in 3851 df-ss 3861 df-br 5032 df-res 5538 |
This theorem is referenced by: (None) |
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