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Mirrors > Home > MPE Home > Th. List > Mathboxes > idresssidinxp | Structured version Visualization version GIF version |
Description: Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) |
Ref | Expression |
---|---|
idresssidinxp | ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 5873 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ I | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ I ) |
3 | idssxp 5911 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
4 | xpss2 5570 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵)) | |
5 | 3, 4 | sstrid 3978 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
6 | 2, 5 | ssind 4209 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3935 ⊆ wss 3936 I cid 5454 × cxp 5548 ↾ cres 5552 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-res 5562 |
This theorem is referenced by: idreseqidinxp 35561 |
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