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Mirrors > Home > MPE Home > Th. List > Mathboxes > brssrres | Structured version Visualization version GIF version |
Description: Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) |
Ref | Expression |
---|---|
brssrres | ⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brres 6007 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 S 𝐶))) | |
2 | brssr 38483 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 S 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
3 | 2 | anbi2d 630 | . 2 ⊢ (𝐶 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵 S 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
4 | 1, 3 | bitrd 279 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 ↾ cres 5691 S cssr 38165 |
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-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-res 5701 df-ssr 38480 |
This theorem is referenced by: br1cnvssrres 38487 |
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