Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > resindm | Structured version Visualization version GIF version |
Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.) |
Ref | Expression |
---|---|
resindm | ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdm 5899 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
2 | 1 | ineq2d 4191 | . 2 ⊢ (Rel 𝐴 → ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ 𝐴)) |
3 | resindi 5871 | . 2 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) | |
4 | incom 4180 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴 ↾ 𝐵)) | |
5 | inres 5873 | . . 3 ⊢ (𝐴 ∩ (𝐴 ↾ 𝐵)) = ((𝐴 ∩ 𝐴) ↾ 𝐵) | |
6 | inidm 4197 | . . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | 6 | reseq1i 5851 | . . 3 ⊢ ((𝐴 ∩ 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
8 | 4, 5, 7 | 3eqtrri 2851 | . 2 ⊢ (𝐴 ↾ 𝐵) = ((𝐴 ↾ 𝐵) ∩ 𝐴) |
9 | 2, 3, 8 | 3eqtr4g 2883 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3937 dom cdm 5557 ↾ cres 5559 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-res 5569 |
This theorem is referenced by: resdmdfsn 5903 resfnfinfin 8806 resfifsupp 8863 poimirlem3 34897 fresin2 41435 limsupvaluz 41996 cncfuni 42176 fourierdlem48 42446 fourierdlem49 42447 fourierdlem113 42511 sssmf 43022 |
Copyright terms: Public domain | W3C validator |