Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3988 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssv 3990 | . . . 4 ⊢ 𝐵 ⊆ V | |
3 | xpss12 5564 | . . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V)) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V) |
5 | sslin 4210 | . . 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 5561 | . 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
8 | 6, 7 | sseqtrri 4003 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 × cxp 5547 ↾ cres 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-opab 5121 df-xp 5555 df-res 5561 |
This theorem is referenced by: ssrnres 6029 idreseqidinxp 35561 refrelsredund4 35861 refrelredund4 35864 |
Copyright terms: Public domain | W3C validator |