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Mirrors > Home > ILE Home > Th. List > ssres2 | 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 4757 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V)) | |
2 | sslin 3376 | . . 3 ⊢ ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) |
4 | df-res 4659 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
5 | df-res 4659 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
6 | 3, 4, 5 | 3sstr4g 3213 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Vcvv 2752 ∩ cin 3143 ⊆ wss 3144 × cxp 4645 ↾ cres 4649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-opab 4083 df-xp 4653 df-res 4659 |
This theorem is referenced by: imass2 5025 resasplitss 5417 fnsnsplitss 5739 1stcof 6192 2ndcof 6193 |
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