Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funresdm1 | 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 |
---|---|
funresdm1 | ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundir 5851 | . 2 ⊢ ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) | |
2 | resdm 5881 | . . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
3 | 2 | adantr 484 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴) |
4 | dmres 5858 | . . . . . 6 ⊢ dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵) | |
5 | simpr 488 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅) | |
6 | 4, 5 | syl5eq 2783 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅) |
7 | relres 5865 | . . . . . 6 ⊢ Rel (𝐵 ↾ dom 𝐴) | |
8 | reldm0 5782 | . . . . . 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 237 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅) |
11 | 3, 10 | uneq12d 4064 | . . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅)) |
12 | un0 4291 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | eqtrdi 2787 | . 2 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴) |
14 | 1, 13 | syl5eq 2783 | 1 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∪ cun 3851 ∩ cin 3852 ∅c0 4223 dom cdm 5536 ↾ cres 5538 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-dm 5546 df-res 5548 |
This theorem is referenced by: fnunres1 30618 |
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