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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 5658 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ I | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ I ) |
3 | idssxp 5697 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
4 | xpss2 5362 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵)) | |
5 | 3, 4 | syl5ss 3838 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
6 | 2, 5 | ssind 4061 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3797 ⊆ wss 3798 I cid 5249 × cxp 5340 ↾ cres 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-res 5354 |
This theorem is referenced by: idreseqidinxp 34622 |
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