Theorem djudm 6840

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 6838	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
2	1	dmeqi 4650	. 2 |- dom ( F ⊔d G ) = dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
3		dmun 4656	. 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 4952	. . . . 5 |- dom ( F o. `' (inl |` dom F ) ) = ( `' `' (inl |` dom F ) " dom F )
5		imacnvcnv 4908	. . . . 5 |- ( `' `' (inl |` dom F ) " dom F ) = ( (inl |` dom F ) " dom F )
6		resima 4758	. . . . . 6 |- ( (inl |` dom F ) " dom F ) = (inl " dom F )
7		df-ima 4464	. . . . . 6 |- (inl " dom F ) = ran (inl |` dom F )
8	6, 7	eqtri 2109	. . . . 5 |- ( (inl |` dom F ) " dom F ) = ran (inl |` dom F )
9	4, 5, 8	3eqtri 2113	. . . 4 |- dom ( F o. `' (inl |` dom F ) ) = ran (inl |` dom F )
10		dmco 4952	. . . . 5 |- dom ( G o. `' (inr |` dom G ) ) = ( `' `' (inr |` dom G ) " dom G )
11		imacnvcnv 4908	. . . . 5 |- ( `' `' (inr |` dom G ) " dom G ) = ( (inr |` dom G ) " dom G )
12		resima 4758	. . . . . 6 |- ( (inr |` dom G ) " dom G ) = (inr " dom G )
13		df-ima 4464	. . . . . 6 |- (inr " dom G ) = ran (inr |` dom G )
14	12, 13	eqtri 2109	. . . . 5 |- ( (inr |` dom G ) " dom G ) = ran (inr |` dom G )
15	10, 11, 14	3eqtri 2113	. . . 4 |- dom ( G o. `' (inr |` dom G ) ) = ran (inr |` dom G )
16	9, 15	uneq12i 3153	. . 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 6812	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
18	16, 17	eqtri 2109	. 2 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) ) = ( dom F ⊔ dom G )
19	2, 3, 18	3eqtri 2113	1 |- dom ( F ⊔d G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1290   u. cun 2998   `'ccnv 4450   dom cdm 4451   ran crn 4452   |` cres 4453   "cima 4454   o. ccom 4455   ⊔ cdju 6784  inlcinl 6791  inrcinr 6792   ⊔d cdjud 6837

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1st 5925  df-2nd 5926  df-1o 6195  df-dju 6785  df-inl 6793  df-inr 6794  df-djud 6838

This theorem is referenced by: (None)