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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 5708 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V)) | |
2 | sslin 4251 | . . 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 5701 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
5 | df-res 5701 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
6 | 3, 4, 5 | 3sstr4g 4041 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 × cxp 5687 ↾ cres 5691 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-opab 5211 df-xp 5695 df-res 5701 |
This theorem is referenced by: imass2 6123 1stcof 8043 2ndcof 8044 tfrlem15 8431 gsum2dlem2 20004 txkgen 23676 funpsstri 35747 eldisjss 38720 resnonrel 43582 mptrcllem 43603 rtrclexi 43611 cnvrcl0 43615 relexpss1d 43695 relexp0a 43706 supcnvlimsup 45696 |
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