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| Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.) | 
| Ref | Expression | 
|---|---|
| resindi | ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpindir 5845 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
| 2 | 1 | ineq2i 4217 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | 
| 3 | inindi 4235 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V))) | |
| 4 | 2, 3 | eqtri 2765 | . 2 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V))) | 
| 5 | df-res 5697 | . 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
| 6 | df-res 5697 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
| 7 | df-res 5697 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
| 8 | 6, 7 | ineq12i 4218 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V))) | 
| 9 | 4, 5, 8 | 3eqtr4i 2775 | 1 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ↾ cres 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 df-res 5697 | 
| This theorem is referenced by: resindm 6048 gsum2dlem2 19989 fnresin 32636 fressupp 32697 disjresdisj 38242 | 
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