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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 6019 | . 2 ⊢ (𝑅 ↾ 𝐶) ⊆ 𝑅 | |
| 2 | 1 | ssbri 5188 | 1 ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ↾ cres 5687 |
| 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-v 3482 df-in 3958 df-ss 3968 df-br 5144 df-res 5697 |
| This theorem is referenced by: (None) |
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