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Mirrors > Home > MPE Home > Th. List > idinxpres | Structured version Visualization version GIF version |
Description: The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 6046) to cartesian product. (Revised by BJ, 23-Dec-2023.) |
Ref | Expression |
---|---|
idinxpres | ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elidinxp 6042 | . . 3 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩) | |
2 | elrid 6044 | . . 3 ⊢ (𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩) | |
3 | 1, 2 | bitr4i 277 | . 2 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵))) |
4 | 3 | eqriv 2722 | 1 ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ∩ cin 3938 ⟨cop 4630 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 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-res 5684 |
This theorem is referenced by: idinxpresid 6046 |
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