Theorem djudm 6990

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 6988	. . 3 |- ( F ⊔d G ) = ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
2	1	dmeqi 4740	. 2 |- dom ( F ⊔d G ) = dom ( ( F o. `' (inl |` dom F ) ) u. ( G o. `' (inr |` dom G ) ) )
3		dmun 4746	. 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 5047	. . . . 5 |- dom ( F o. `' (inl |` dom F ) ) = ( `' `' (inl |` dom F ) " dom F )
5		imacnvcnv 5003	. . . . 5 |- ( `' `' (inl |` dom F ) " dom F ) = ( (inl |` dom F ) " dom F )
6		resima 4852	. . . . . 6 |- ( (inl |` dom F ) " dom F ) = (inl " dom F )
7		df-ima 4552	. . . . . 6 |- (inl " dom F ) = ran (inl |` dom F )
8	6, 7	eqtri 2160	. . . . 5 |- ( (inl |` dom F ) " dom F ) = ran (inl |` dom F )
9	4, 5, 8	3eqtri 2164	. . . 4 |- dom ( F o. `' (inl |` dom F ) ) = ran (inl |` dom F )
10		dmco 5047	. . . . 5 |- dom ( G o. `' (inr |` dom G ) ) = ( `' `' (inr |` dom G ) " dom G )
11		imacnvcnv 5003	. . . . 5 |- ( `' `' (inr |` dom G ) " dom G ) = ( (inr |` dom G ) " dom G )
12		resima 4852	. . . . . 6 |- ( (inr |` dom G ) " dom G ) = (inr " dom G )
13		df-ima 4552	. . . . . 6 |- (inr " dom G ) = ran (inr |` dom G )
14	12, 13	eqtri 2160	. . . . 5 |- ( (inr |` dom G ) " dom G ) = ran (inr |` dom G )
15	10, 11, 14	3eqtri 2164	. . . 4 |- dom ( G o. `' (inr |` dom G ) ) = ran (inr |` dom G )
16	9, 15	uneq12i 3228	. . 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 6951	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
18	16, 17	eqtri 2160	. 2 |- ( dom ( F o. `' (inl |` dom F ) ) u. dom ( G o. `' (inr |` dom G ) ) ) = ( dom F ⊔ dom G )
19	2, 3, 18	3eqtri 2164	1 |- dom ( F ⊔d G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1331   u. cun 3069   `'ccnv 4538   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542   o. ccom 4543   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931   ⊔d cdjud 6987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-djud 6988

This theorem is referenced by: (None)