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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 4242 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V))) | |
| 2 | df-res 5697 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
| 3 | df-res 5697 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
| 4 | 1, 2, 3 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ↾ 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-res 5697 |
| This theorem is referenced by: imass1 6119 marypha1lem 9473 sspg 30747 ssps 30749 sspn 30755 |
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