Theorem djudm 7164

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.)

Assertion

Ref	Expression
djudm	⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Proof of Theorem djudm

Step	Hyp	Ref	Expression
1		df-djud 7162	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
2	1	dmeqi 4863	. 2 ⊢ dom (𝐹 ⊔d 𝐺) = dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
3		dmun 4869	. 2 ⊢ dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
4		dmco 5174	. . . . 5 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = (◡◡(inl ↾ dom 𝐹) “ dom 𝐹)
5		imacnvcnv 5130	. . . . 5 ⊢ (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹)
6		resima 4975	. . . . . 6 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹)
7		df-ima 4672	. . . . . 6 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
8	6, 7	eqtri 2214	. . . . 5 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹)
9	4, 5, 8	3eqtri 2218	. . . 4 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹)
10		dmco 5174	. . . . 5 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = (◡◡(inr ↾ dom 𝐺) “ dom 𝐺)
11		imacnvcnv 5130	. . . . 5 ⊢ (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺)
12		resima 4975	. . . . . 6 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺)
13		df-ima 4672	. . . . . 6 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
14	12, 13	eqtri 2214	. . . . 5 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺)
15	10, 11, 14	3eqtri 2218	. . . 4 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺)
16	9, 15	uneq12i 3311	. . 3 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
17		djuunr 7125	. . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
18	16, 17	eqtri 2214	. 2 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺)
19	2, 3, 18	3eqtri 2218	1 ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Syntax hints:   = wceq 1364   ∪ cun 3151  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   ↾ cres 4661   “ cima 4662   ∘ ccom 4663   ⊔ cdju 7096  inlcinl 7104  inrcinr 7105   ⊔_d cdjud 7161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-dju 7097  df-inl 7106  df-inr 7107  df-djud 7162

This theorem is referenced by: (None)