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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 5878 | . 2 ⊢ (𝑅 ↾ 𝐶) ⊆ 𝑅 | |
2 | 1 | ssbri 5111 | 1 ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-br 5067 df-res 5567 |
This theorem is referenced by: (None) |
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