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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 6037) to cartesian product. (Revised by BJ, 23-Dec-2023.) |
Ref | Expression |
---|---|
idinxpres | ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elidinxp 6033 | . . 3 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩) | |
2 | elrid 6035 | . . 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 2728 | 1 ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∃wrex 3069 ∩ cin 3943 ⟨cop 4628 I cid 5566 × cxp 5667 ↾ cres 5671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-res 5681 |
This theorem is referenced by: idinxpresid 6037 |
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