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Mathbox for Peter Mazsa |
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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 6004 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ I | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ I ) |
3 | idssxp 6046 | . . 3 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
4 | xpss2 5692 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵)) | |
5 | 3, 4 | sstrid 3989 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
6 | 2, 5 | ssind 4228 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3943 ⊆ wss 3944 I cid 5569 × cxp 5670 ↾ cres 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-res 5684 |
This theorem is referenced by: idreseqidinxp 37704 |
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