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Theorem djudm 7135
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 7133 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
21dmeqi 4846 . 2  |-  dom  ( F ⊔d  G )  =  dom  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) )
3 dmun 4852 . 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 5155 . . . . 5  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ( `' `' (inl  |`  dom  F )
" dom  F )
5 imacnvcnv 5111 . . . . 5  |-  ( `' `' (inl  |`  dom  F
) " dom  F
)  =  ( (inl  |`  dom  F ) " dom  F )
6 resima 4958 . . . . . 6  |-  ( (inl  |`  dom  F ) " dom  F )  =  (inl " dom  F )
7 df-ima 4657 . . . . . 6  |-  (inl " dom  F )  =  ran  (inl  |`  dom  F )
86, 7eqtri 2210 . . . . 5  |-  ( (inl  |`  dom  F ) " dom  F )  =  ran  (inl  |`  dom  F )
94, 5, 83eqtri 2214 . . . 4  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ran  (inl  |`  dom  F
)
10 dmco 5155 . . . . 5  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ( `' `' (inr  |`  dom  G )
" dom  G )
11 imacnvcnv 5111 . . . . 5  |-  ( `' `' (inr  |`  dom  G
) " dom  G
)  =  ( (inr  |`  dom  G ) " dom  G )
12 resima 4958 . . . . . 6  |-  ( (inr  |`  dom  G ) " dom  G )  =  (inr " dom  G )
13 df-ima 4657 . . . . . 6  |-  (inr " dom  G )  =  ran  (inr  |`  dom  G )
1412, 13eqtri 2210 . . . . 5  |-  ( (inr  |`  dom  G ) " dom  G )  =  ran  (inr  |`  dom  G )
1510, 11, 143eqtri 2214 . . . 4  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ran  (inr  |`  dom  G
)
169, 15uneq12i 3302 . . 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 7096 . . 3  |-  ( ran  (inl  |`  dom  F )  u.  ran  (inr  |`  dom  G
) )  =  ( dom  F dom  G )
1816, 17eqtri 2210 . 2  |-  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  u.  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  ( dom 
F dom  G )
192, 3, 183eqtri 2214 1  |-  dom  ( F ⊔d  G )  =  ( dom  F dom  G )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    u. cun 3142   `'ccnv 4643   dom cdm 4644   ran crn 4645    |` cres 4646   "cima 4647    o. ccom 4648   ⊔ cdju 7067  inlcinl 7075  inrcinr 7076   ⊔d cdjud 7132
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-dju 7068  df-inl 7077  df-inr 7078  df-djud 7133
This theorem is referenced by: (None)
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