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Theorem djudm 7097
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 7095 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
21dmeqi 4823 . 2  |-  dom  ( F ⊔d  G )  =  dom  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) )
3 dmun 4829 . 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 5132 . . . . 5  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ( `' `' (inl  |`  dom  F )
" dom  F )
5 imacnvcnv 5088 . . . . 5  |-  ( `' `' (inl  |`  dom  F
) " dom  F
)  =  ( (inl  |`  dom  F ) " dom  F )
6 resima 4935 . . . . . 6  |-  ( (inl  |`  dom  F ) " dom  F )  =  (inl " dom  F )
7 df-ima 4635 . . . . . 6  |-  (inl " dom  F )  =  ran  (inl  |`  dom  F )
86, 7eqtri 2198 . . . . 5  |-  ( (inl  |`  dom  F ) " dom  F )  =  ran  (inl  |`  dom  F )
94, 5, 83eqtri 2202 . . . 4  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ran  (inl  |`  dom  F
)
10 dmco 5132 . . . . 5  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ( `' `' (inr  |`  dom  G )
" dom  G )
11 imacnvcnv 5088 . . . . 5  |-  ( `' `' (inr  |`  dom  G
) " dom  G
)  =  ( (inr  |`  dom  G ) " dom  G )
12 resima 4935 . . . . . 6  |-  ( (inr  |`  dom  G ) " dom  G )  =  (inr " dom  G )
13 df-ima 4635 . . . . . 6  |-  (inr " dom  G )  =  ran  (inr  |`  dom  G )
1412, 13eqtri 2198 . . . . 5  |-  ( (inr  |`  dom  G ) " dom  G )  =  ran  (inr  |`  dom  G )
1510, 11, 143eqtri 2202 . . . 4  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ran  (inr  |`  dom  G
)
169, 15uneq12i 3287 . . 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 7058 . . 3  |-  ( ran  (inl  |`  dom  F )  u.  ran  (inr  |`  dom  G
) )  =  ( dom  F dom  G )
1816, 17eqtri 2198 . 2  |-  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  u.  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  ( dom 
F dom  G )
192, 3, 183eqtri 2202 1  |-  dom  ( F ⊔d  G )  =  ( dom  F dom  G )
Colors of variables: wff set class
Syntax hints:    = wceq 1353    u. cun 3127   `'ccnv 4621   dom cdm 4622   ran crn 4623    |` cres 4624   "cima 4625    o. ccom 4626   ⊔ cdju 7029  inlcinl 7037  inrcinr 7038   ⊔d cdjud 7094
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-1st 6134  df-2nd 6135  df-1o 6410  df-dju 7030  df-inl 7039  df-inr 7040  df-djud 7095
This theorem is referenced by: (None)
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