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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 3849 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssv 3851 | . . . 4 ⊢ 𝐵 ⊆ V | |
3 | xpss12 5358 | . . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V)) | |
4 | 1, 2, 3 | mp2an 685 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V) |
5 | sslin 4064 | . . 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 5355 | . 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
8 | 6, 7 | sseqtr4i 3864 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3415 ∩ cin 3798 ⊆ wss 3799 × cxp 5341 ↾ cres 5345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-v 3417 df-in 3806 df-ss 3813 df-opab 4937 df-xp 5349 df-res 5355 |
This theorem is referenced by: ssrnres 5814 idreseqidinxp 34630 refrelsred4 34922 refrelred4 34925 |
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