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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 3943 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssv 3945 | . . . 4 ⊢ 𝐵 ⊆ V | |
3 | xpss12 5604 | . . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V)) | |
4 | 1, 2, 3 | mp2an 689 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V) |
5 | sslin 4168 | . . 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 5601 | . 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
8 | 6, 7 | sseqtrri 3958 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 × cxp 5587 ↾ cres 5591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-in 3894 df-ss 3904 df-opab 5137 df-xp 5595 df-res 5601 |
This theorem is referenced by: ssrnres 6081 idreseqidinxp 36445 refrelsredund4 36745 refrelredund4 36748 |
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