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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 7049 | . . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
2 | 1 | dmeqi 4789 | . 2 ⊢ dom (𝐹 ⊔d 𝐺) = dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) |
3 | dmun 4795 | . 2 ⊢ dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
4 | dmco 5096 | . . . . 5 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) | |
5 | imacnvcnv 5052 | . . . . 5 ⊢ (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹) | |
6 | resima 4901 | . . . . . 6 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹) | |
7 | df-ima 4601 | . . . . . 6 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹) | |
8 | 6, 7 | eqtri 2178 | . . . . 5 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹) |
9 | 4, 5, 8 | 3eqtri 2182 | . . . 4 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹) |
10 | dmco 5096 | . . . . 5 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) | |
11 | imacnvcnv 5052 | . . . . 5 ⊢ (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺) | |
12 | resima 4901 | . . . . . 6 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺) | |
13 | df-ima 4601 | . . . . . 6 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺) | |
14 | 12, 13 | eqtri 2178 | . . . . 5 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺) |
15 | 10, 11, 14 | 3eqtri 2182 | . . . 4 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺) |
16 | 9, 15 | uneq12i 3260 | . . 3 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) |
17 | djuunr 7012 | . . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺) | |
18 | 16, 17 | eqtri 2178 | . 2 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺) |
19 | 2, 3, 18 | 3eqtri 2182 | 1 ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∪ cun 3100 ◡ccnv 4587 dom cdm 4588 ran crn 4589 ↾ cres 4590 “ cima 4591 ∘ ccom 4592 ⊔ cdju 6983 inlcinl 6991 inrcinr 6992 ⊔d cdjud 7048 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-1st 6090 df-2nd 6091 df-1o 6365 df-dju 6984 df-inl 6993 df-inr 6994 df-djud 7049 |
This theorem is referenced by: (None) |
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