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| Mirrors > Home > MPE Home > Th. List > inxpssres | Structured version Visualization version GIF version | ||
| Description: Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| inxpssres | ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssid 4005 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
| 2 | ssv 4007 | . . . 4 ⊢ 𝐵 ⊆ V | |
| 3 | xpss12 5699 | . . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V)) | |
| 4 | 1, 2, 3 | mp2an 692 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V) | 
| 5 | sslin 4242 | . . 3 ⊢ ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V))) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V)) | 
| 7 | df-res 5696 | . 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
| 8 | 6, 7 | sseqtrri 4032 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 × cxp 5682 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 df-opab 5205 df-xp 5690 df-res 5696 | 
| This theorem is referenced by: ssrnres 6197 idreseqidinxp 38311 refrelsredund4 38634 refrelredund4 38637 | 
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