Theorem djudm 7171

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 7169	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
2	1	dmeqi 4867	. 2 |- dom ( F ⊔d G ) = dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
3		dmun 4873	. 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 5178	. . . . 5 |- dom ( F o. `' (inl |` dom F ) ) = ( `' `' (inl |` dom F ) " dom F )
5		imacnvcnv 5134	. . . . 5 |- ( `' `' (inl |` dom F ) " dom F ) = ( (inl |` dom F ) " dom F )
6		resima 4979	. . . . . 6 |- ( (inl |` dom F ) " dom F ) = (inl " dom F )
7		df-ima 4676	. . . . . 6 |- (inl " dom F ) = ran (inl |` dom F )
8	6, 7	eqtri 2217	. . . . 5 |- ( (inl |` dom F ) " dom F ) = ran (inl |` dom F )
9	4, 5, 8	3eqtri 2221	. . . 4 |- dom ( F o. `' (inl |` dom F ) ) = ran (inl |` dom F )
10		dmco 5178	. . . . 5 |- dom ( G o. `' (inr |` dom G ) ) = ( `' `' (inr |` dom G ) " dom G )
11		imacnvcnv 5134	. . . . 5 |- ( `' `' (inr |` dom G ) " dom G ) = ( (inr |` dom G ) " dom G )
12		resima 4979	. . . . . 6 |- ( (inr |` dom G ) " dom G ) = (inr " dom G )
13		df-ima 4676	. . . . . 6 |- (inr " dom G ) = ran (inr |` dom G )
14	12, 13	eqtri 2217	. . . . 5 |- ( (inr |` dom G ) " dom G ) = ran (inr |` dom G )
15	10, 11, 14	3eqtri 2221	. . . 4 |- dom ( G o. `' (inr |` dom G ) ) = ran (inr |` dom G )
16	9, 15	uneq12i 3315	. . 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 7132	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
18	16, 17	eqtri 2217	. 2 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) ) = ( dom F ⊔ dom G )
19	2, 3, 18	3eqtri 2221	1 |- dom ( F ⊔d G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1364   u. cun 3155   `'ccnv 4662   dom cdm 4663   ran crn 4664   |` cres 4665   "cima 4666   o. ccom 4667   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112   ⊔d cdjud 7168

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114  df-djud 7169

This theorem is referenced by: (None)