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Theorem djudm 6958
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 6956 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
21dmeqi 4710 . 2  |-  dom  ( F ⊔d  G )  =  dom  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) )
3 dmun 4716 . 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 5017 . . . . 5  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ( `' `' (inl  |`  dom  F )
" dom  F )
5 imacnvcnv 4973 . . . . 5  |-  ( `' `' (inl  |`  dom  F
) " dom  F
)  =  ( (inl  |`  dom  F ) " dom  F )
6 resima 4822 . . . . . 6  |-  ( (inl  |`  dom  F ) " dom  F )  =  (inl " dom  F )
7 df-ima 4522 . . . . . 6  |-  (inl " dom  F )  =  ran  (inl  |`  dom  F )
86, 7eqtri 2138 . . . . 5  |-  ( (inl  |`  dom  F ) " dom  F )  =  ran  (inl  |`  dom  F )
94, 5, 83eqtri 2142 . . . 4  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) )  =  ran  (inl  |`  dom  F
)
10 dmco 5017 . . . . 5  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ( `' `' (inr  |`  dom  G )
" dom  G )
11 imacnvcnv 4973 . . . . 5  |-  ( `' `' (inr  |`  dom  G
) " dom  G
)  =  ( (inr  |`  dom  G ) " dom  G )
12 resima 4822 . . . . . 6  |-  ( (inr  |`  dom  G ) " dom  G )  =  (inr " dom  G )
13 df-ima 4522 . . . . . 6  |-  (inr " dom  G )  =  ran  (inr  |`  dom  G )
1412, 13eqtri 2138 . . . . 5  |-  ( (inr  |`  dom  G ) " dom  G )  =  ran  (inr  |`  dom  G )
1510, 11, 143eqtri 2142 . . . 4  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) )  =  ran  (inr  |`  dom  G
)
169, 15uneq12i 3198 . . 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 6919 . . 3  |-  ( ran  (inl  |`  dom  F )  u.  ran  (inr  |`  dom  G
) )  =  ( dom  F dom  G )
1816, 17eqtri 2138 . 2  |-  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  u.  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  ( dom 
F dom  G )
192, 3, 183eqtri 2142 1  |-  dom  ( F ⊔d  G )  =  ( dom  F dom  G )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    u. cun 3039   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511   "cima 4512    o. ccom 4513   ⊔ cdju 6890  inlcinl 6898  inrcinr 6899   ⊔d cdjud 6955
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901  df-djud 6956
This theorem is referenced by: (None)
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