Theorem djudm 7272

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 7270	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
2	1	dmeqi 4924	. 2 |- dom ( F ⊔d G ) = dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
3		dmun 4930	. 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 5237	. . . . 5 |- dom ( F o. `' (inl |` dom F ) ) = ( `' `' (inl |` dom F ) " dom F )
5		imacnvcnv 5193	. . . . 5 |- ( `' `' (inl |` dom F ) " dom F ) = ( (inl |` dom F ) " dom F )
6		resima 5038	. . . . . 6 |- ( (inl |` dom F ) " dom F ) = (inl " dom F )
7		df-ima 4732	. . . . . 6 |- (inl " dom F ) = ran (inl |` dom F )
8	6, 7	eqtri 2250	. . . . 5 |- ( (inl |` dom F ) " dom F ) = ran (inl |` dom F )
9	4, 5, 8	3eqtri 2254	. . . 4 |- dom ( F o. `' (inl |` dom F ) ) = ran (inl |` dom F )
10		dmco 5237	. . . . 5 |- dom ( G o. `' (inr |` dom G ) ) = ( `' `' (inr |` dom G ) " dom G )
11		imacnvcnv 5193	. . . . 5 |- ( `' `' (inr |` dom G ) " dom G ) = ( (inr |` dom G ) " dom G )
12		resima 5038	. . . . . 6 |- ( (inr |` dom G ) " dom G ) = (inr " dom G )
13		df-ima 4732	. . . . . 6 |- (inr " dom G ) = ran (inr |` dom G )
14	12, 13	eqtri 2250	. . . . 5 |- ( (inr |` dom G ) " dom G ) = ran (inr |` dom G )
15	10, 11, 14	3eqtri 2254	. . . 4 |- dom ( G o. `' (inr |` dom G ) ) = ran (inr |` dom G )
16	9, 15	uneq12i 3356	. . 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 7233	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
18	16, 17	eqtri 2250	. 2 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) ) = ( dom F ⊔ dom G )
19	2, 3, 18	3eqtri 2254	1 |- dom ( F ⊔d G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1395   u. cun 3195   `'ccnv 4718   dom cdm 4719   ran crn 4720   |` cres 4721   "cima 4722   o. ccom 4723   ⊔ cdju 7204  inlcinl 7212  inrcinr 7213   ⊔d cdjud 7269

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215  df-djud 7270

This theorem is referenced by: (None)