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Mirrors > Home > MPE Home > Th. List > ssres2 | Structured version Visualization version GIF version |
Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssres2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss1 5151 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V)) | |
2 | sslin 3801 | . . 3 ⊢ ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) |
4 | df-res 5050 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
5 | df-res 5050 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
6 | 3, 4, 5 | 3sstr4g 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 × cxp 5036 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-opab 4644 df-xp 5044 df-res 5050 |
This theorem is referenced by: imass2 5420 1stcof 7087 2ndcof 7088 tfrlem15 7375 gsum2dlem2 18193 txkgen 21265 funpsstri 30909 resnonrel 36917 mptrcllem 36939 rtrclexi 36947 cnvrcl0 36951 relexpss1d 37016 relexp0a 37027 |
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