| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > relresdm1 | Structured version Visualization version GIF version | ||
| Description: Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
| Ref | Expression |
|---|---|
| relresdm1 | ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | resundir 6012 | . 2 ⊢ ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) | |
| 2 | resdm 6044 | . . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴) |
| 4 | dmres 6030 | . . . . . 6 ⊢ dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵) | |
| 5 | simpr 484 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅) | |
| 6 | 4, 5 | eqtrid 2789 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅) |
| 7 | relres 6023 | . . . . . 6 ⊢ Rel (𝐵 ↾ dom 𝐴) | |
| 8 | reldm0 5938 | . . . . . 6 ⊢ (Rel (𝐵 ↾ dom 𝐴) → ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅)) | |
| 9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅) |
| 10 | 6, 9 | sylibr 234 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅) |
| 11 | 3, 10 | uneq12d 4169 | . . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅)) |
| 12 | un0 4394 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 13 | 11, 12 | eqtrdi 2793 | . 2 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴) |
| 14 | 1, 13 | eqtrid 2789 | 1 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 |
| 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 |
| This theorem is referenced by: fnunres1 6680 |
| Copyright terms: Public domain | W3C validator |