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Mirrors > Home > MPE Home > Th. List > inres | Structured version Visualization version GIF version |
Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.) |
Ref | Expression |
---|---|
inres | ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4177 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 5643 | . 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) | |
3 | df-res 5643 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
4 | 3 | ineq2i 4167 | . 2 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V))) |
5 | 1, 2, 4 | 3eqtr4ri 2775 | 1 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3443 ∩ cin 3907 × cxp 5629 ↾ cres 5633 |
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 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-in 3915 df-res 5643 |
This theorem is referenced by: resindm 5984 fninfp 7116 symgcom2 31777 inres2 36635 xrnres2 36797 br1cossinres 36841 |
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