Theorem djudm 7166

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 7164	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
2	1	dmeqi 4864	. 2 |- dom ( F ⊔d G ) = dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
3		dmun 4870	. 2 |- dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) ) = ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) )
4		dmco 5175	. . . . 5 |- dom ( F o. `' (inl |` dom F ) ) = ( `' `' (inl |` dom F ) " dom F )
5		imacnvcnv 5131	. . . . 5 |- ( `' `' (inl |` dom F ) " dom F ) = ( (inl |` dom F ) " dom F )
6		resima 4976	. . . . . 6 |- ( (inl |` dom F ) " dom F ) = (inl " dom F )
7		df-ima 4673	. . . . . 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 o. `' (inl |` dom F ) ) = ran (inl |` dom F )
10		dmco 5175	. . . . 5 |- dom ( G o. `' (inr |` dom G ) ) = ( `' `' (inr |` dom G ) " dom G )
11		imacnvcnv 5131	. . . . 5 |- ( `' `' (inr |` dom G ) " dom G ) = ( (inr |` dom G ) " dom G )
12		resima 4976	. . . . . 6 |- ( (inr |` dom G ) " dom G ) = (inr " dom G )
13		df-ima 4673	. . . . . 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 o. `' (inr |` dom G ) ) = ran (inr |` dom G )
16	9, 15	uneq12i 3312	. . 3 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) ) = ( ran (inl |` dom F ) u. ran (inr |` dom G ) )
17		djuunr 7127	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
18	16, 17	eqtri 2214	. 2 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (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   u. cun 3152   `'ccnv 4659   dom cdm 4660   ran crn 4661   |` cres 4662   "cima 4663   o. ccom 4664   ⊔ cdju 7098  inlcinl 7106  inrcinr 7107   ⊔d cdjud 7163

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 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

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 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109  df-djud 7164

This theorem is referenced by: (None)