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Theorem casedm 6833
Description: The domain of the "case" construction is the disjoint union of the domains. TODO (although less important):  |-  ran case ( F ,  G )  =  ( ran  F  u.  ran  G ). (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
casedm  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )

Proof of Theorem casedm
StepHypRef Expression
1 df-case 6831 . . 3  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21dmeqi 4652 . 2  |-  dom case ( F ,  G )  =  dom  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) )
3 dmun 4658 . 2  |-  dom  (
( F  o.  `'inl )  u.  ( G  o.  `'inr ) )  =  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )
4 dmco 4954 . . . . 5  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
5 imacnvcnv 4910 . . . . 5  |-  ( `' `'inl " dom  F )  =  (inl " dom  F )
6 df-ima 4467 . . . . 5  |-  (inl " dom  F )  =  ran  (inl  |`  dom  F )
74, 5, 63eqtri 2113 . . . 4  |-  dom  ( F  o.  `'inl )  =  ran  (inl  |`  dom  F
)
8 dmco 4954 . . . . 5  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
9 imacnvcnv 4910 . . . . 5  |-  ( `' `'inr " dom  G )  =  (inr " dom  G )
10 df-ima 4467 . . . . 5  |-  (inr " dom  G )  =  ran  (inr  |`  dom  G )
118, 9, 103eqtri 2113 . . . 4  |-  dom  ( G  o.  `'inr )  =  ran  (inr  |`  dom  G
)
127, 11uneq12i 3155 . . 3  |-  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )  =  ( ran  (inl  |` 
dom  F )  u. 
ran  (inr  |`  dom  G
) )
13 djuunr 6814 . . 3  |-  ( ran  (inl  |`  dom  F )  u.  ran  (inr  |`  dom  G
) )  =  ( dom  F dom  G )
1412, 13eqtri 2109 . 2  |-  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )  =  ( dom  F dom 
G )
152, 3, 143eqtri 2113 1  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )
Colors of variables: wff set class
Syntax hints:    = wceq 1290    u. cun 3000   `'ccnv 4453   dom cdm 4454   ran crn 4455    |` cres 4456   "cima 4457    o. ccom 4458   ⊔ cdju 6786  inlcinl 6793  inrcinr 6794  casecdjucase 6830
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 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271
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 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-iord 4204  df-on 4206  df-suc 4209  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-1st 5927  df-2nd 5928  df-1o 6197  df-dju 6787  df-inl 6795  df-inr 6796  df-case 6831
This theorem is referenced by:  casef  6835
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