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Mirrors > Home > MPE Home > Th. List > idinxpresid | Structured version Visualization version GIF version |
Description: The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.) |
Ref | Expression |
---|---|
idinxpresid | ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idinxpres 6035 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴 ∩ 𝐴)) | |
2 | inidm 4213 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
3 | 2 | reseq2i 5969 | . 2 ⊢ ( I ↾ (𝐴 ∩ 𝐴)) = ( I ↾ 𝐴) |
4 | 1, 3 | eqtri 2759 | 1 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∩ cin 3942 I cid 5565 × cxp 5666 ↾ cres 5670 |
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 5291 ax-nul 5298 ax-pr 5419 |
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 3474 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5141 df-opab 5203 df-id 5566 df-xp 5674 df-rel 5675 df-res 5680 |
This theorem is referenced by: idssxp 6037 bj-diagval2 35846 idinxpssinxp2 36978 |
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