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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3624 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssv 3625 | . . . 4 ⊢ 𝐵 ⊆ V | |
3 | xpss12 5225 | . . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V)) | |
4 | 1, 2, 3 | mp2an 708 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V) |
5 | sslin 3839 | . . 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 5126 | . 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
8 | 6, 7 | sseqtr4i 3638 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-res 5126 |
This theorem is referenced by: idreseqidinxp 34080 idinxpres 34088 |
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