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Mirrors > Home > MPE Home > Th. List > ssres | Structured version Visualization version GIF version |
Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
ssres | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4249 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 5700 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
3 | df-res 5700 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
4 | 1, 2, 3 | 3sstr4g 4040 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 × cxp 5686 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-in 3969 df-ss 3979 df-res 5700 |
This theorem is referenced by: imass1 6121 marypha1lem 9470 sspg 30756 ssps 30758 sspn 30764 |
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