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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 6047) to cartesian product. (Revised by BJ, 23-Dec-2023.) |
Ref | Expression |
---|---|
idinxpres | ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elidinxp 6043 | . . 3 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = 〈𝑦, 𝑦〉) | |
2 | elrid 6045 | . . 3 ⊢ (𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = 〈𝑦, 𝑦〉) | |
3 | 1, 2 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵))) |
4 | 3 | eqriv 2728 | 1 ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ∩ cin 3947 〈cop 4634 I cid 5573 × cxp 5674 ↾ cres 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-res 5688 |
This theorem is referenced by: idinxpresid 6047 |
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