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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 4209 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 5561 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
3 | df-res 5561 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
4 | 1, 2, 3 | 3sstr4g 4011 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 × cxp 5547 ↾ cres 5551 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-res 5561 |
This theorem is referenced by: imass1 5958 marypha1lem 8891 sspg 28499 ssps 28501 sspn 28507 |
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