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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 5704 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V)) | |
| 2 | sslin 4243 | . . 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 5697 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
| 5 | df-res 5697 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
| 6 | 3, 4, 5 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
| 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-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: imass2 6120 1stcof 8044 2ndcof 8045 tfrlem15 8432 gsum2dlem2 19989 txkgen 23660 funpsstri 35766 eldisjss 38739 resnonrel 43605 mptrcllem 43626 rtrclexi 43634 cnvrcl0 43638 relexpss1d 43718 relexp0a 43729 supcnvlimsup 45755 |
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