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 5861 | . 2 ⊢ ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) | |
2 | resdm 5890 | . . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
3 | 2 | adantr 481 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴) |
4 | dmres 5868 | . . . . . 6 ⊢ dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵) | |
5 | simpr 485 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅) | |
6 | 4, 5 | syl5eq 2865 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅) |
7 | relres 5875 | . . . . . 6 ⊢ Rel (𝐵 ↾ dom 𝐴) | |
8 | reldm0 5791 | . . . . . 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 235 | . . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅) |
11 | 3, 10 | uneq12d 4137 | . . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅)) |
12 | un0 4341 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
13 | 11, 12 | syl6eq 2869 | . 2 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴) |
14 | 1, 13 | syl5eq 2865 | 1 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dm 5558 df-res 5560 |
This theorem is referenced by: fnunres1 30284 |
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