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Mirrors > Home > ILE Home > Th. List > djudm | GIF version |
Description: The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
djudm | ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-djud 6988 | . . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
2 | 1 | dmeqi 4740 | . 2 ⊢ dom (𝐹 ⊔d 𝐺) = dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) |
3 | dmun 4746 | . 2 ⊢ dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
4 | dmco 5047 | . . . . 5 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) | |
5 | imacnvcnv 5003 | . . . . 5 ⊢ (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹) | |
6 | resima 4852 | . . . . . 6 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹) | |
7 | df-ima 4552 | . . . . . 6 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹) | |
8 | 6, 7 | eqtri 2160 | . . . . 5 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹) |
9 | 4, 5, 8 | 3eqtri 2164 | . . . 4 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹) |
10 | dmco 5047 | . . . . 5 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) | |
11 | imacnvcnv 5003 | . . . . 5 ⊢ (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺) | |
12 | resima 4852 | . . . . . 6 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺) | |
13 | df-ima 4552 | . . . . . 6 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺) | |
14 | 12, 13 | eqtri 2160 | . . . . 5 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺) |
15 | 10, 11, 14 | 3eqtri 2164 | . . . 4 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺) |
16 | 9, 15 | uneq12i 3228 | . . 3 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) |
17 | djuunr 6951 | . . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺) | |
18 | 16, 17 | eqtri 2160 | . 2 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺) |
19 | 2, 3, 18 | 3eqtri 2164 | 1 ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∪ cun 3069 ◡ccnv 4538 dom cdm 4539 ran crn 4540 ↾ cres 4541 “ cima 4542 ∘ ccom 4543 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 ⊔d cdjud 6987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 df-djud 6988 |
This theorem is referenced by: (None) |
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