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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
StepHypRef Expression
1 df-djud 6838 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
21dmeqi 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 )
86, 7eqtri 2109 . . . . 5  |-  ( (inl  |`  dom  F ) " dom  F )  =  ran  (inl  |`  dom  F )
94, 5, 83eqtri 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 )
1412, 13eqtri 2109 . . . . 5  |-  ( (inr  |`  dom  G ) " dom  G )  =  ran  (inr  |`  dom  G )
1510, 11, 143eqtri 2113 . . . 4  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ran  (inr  |`  dom  G
)
169, 15uneq12i 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 )
1816, 17eqtri 2109 . 2  |-  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  u.  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  ( dom 
F dom  G )
192, 3, 183eqtri 2113 1  |-  dom  ( F ⊔d  G )  =  ( dom  F dom  G )
Colors of variables: wff set class
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)
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