ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  casedm Unicode version

Theorem casedm 7115
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 7113 . . 3  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21dmeqi 4846 . 2  |-  dom case ( F ,  G )  =  dom  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) )
3 dmun 4852 . 2  |-  dom  (
( F  o.  `'inl )  u.  ( G  o.  `'inr ) )  =  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )
4 dmco 5155 . . . . 5  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
5 imacnvcnv 5111 . . . . 5  |-  ( `' `'inl " dom  F )  =  (inl " dom  F )
6 df-ima 4657 . . . . 5  |-  (inl " dom  F )  =  ran  (inl  |`  dom  F )
74, 5, 63eqtri 2214 . . . 4  |-  dom  ( F  o.  `'inl )  =  ran  (inl  |`  dom  F
)
8 dmco 5155 . . . . 5  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
9 imacnvcnv 5111 . . . . 5  |-  ( `' `'inr " dom  G )  =  (inr " dom  G )
10 df-ima 4657 . . . . 5  |-  (inr " dom  G )  =  ran  (inr  |`  dom  G )
118, 9, 103eqtri 2214 . . . 4  |-  dom  ( G  o.  `'inr )  =  ran  (inr  |`  dom  G
)
127, 11uneq12i 3302 . . 3  |-  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )  =  ( ran  (inl  |` 
dom  F )  u. 
ran  (inr  |`  dom  G
) )
13 djuunr 7095 . . 3  |-  ( ran  (inl  |`  dom  F )  u.  ran  (inr  |`  dom  G
) )  =  ( dom  F dom  G )
1412, 13eqtri 2210 . 2  |-  ( dom  ( F  o.  `'inl )  u.  dom  ( G  o.  `'inr ) )  =  ( dom  F dom 
G )
152, 3, 143eqtri 2214 1  |-  dom case ( F ,  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 7066  inlcinl 7074  inrcinr 7075  casecdjucase 7112
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 6165  df-2nd 6166  df-1o 6441  df-dju 7067  df-inl 7076  df-inr 7077  df-case 7113
This theorem is referenced by:  casef  7117
  Copyright terms: Public domain W3C validator