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Mirrors > Home > MPE Home > Th. List > Mathboxes > resisresindm | Structured version Visualization version GIF version |
Description: The restriction of a relation by a set 𝐵 is identical with the restriction by the intersection of 𝐵 with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.) |
Ref | Expression |
---|---|
resisresindm | ⊢ (Rel 𝐹 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdm 5361 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
2 | 1 | eqcomd 2616 | . . 3 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹)) |
3 | 2 | ineq2d 3776 | . 2 ⊢ (Rel 𝐹 → ((𝐹 ↾ 𝐵) ∩ 𝐹) = ((𝐹 ↾ 𝐵) ∩ (𝐹 ↾ dom 𝐹))) |
4 | resss 5342 | . . . 4 ⊢ (𝐹 ↾ 𝐵) ⊆ 𝐹 | |
5 | df-ss 3554 | . . . 4 ⊢ ((𝐹 ↾ 𝐵) ⊆ 𝐹 ↔ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵)) | |
6 | 4, 5 | mpbi 219 | . . 3 ⊢ ((𝐹 ↾ 𝐵) ∩ 𝐹) = (𝐹 ↾ 𝐵) |
7 | 6 | eqcomi 2619 | . 2 ⊢ (𝐹 ↾ 𝐵) = ((𝐹 ↾ 𝐵) ∩ 𝐹) |
8 | resindi 5332 | . 2 ⊢ (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = ((𝐹 ↾ 𝐵) ∩ (𝐹 ↾ dom 𝐹)) | |
9 | 3, 7, 8 | 3eqtr4g 2669 | 1 ⊢ (Rel 𝐹 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-res 5050 |
This theorem is referenced by: resfnfinfin 40339 |
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