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

Step	Hyp	Ref	Expression
1		df-case 7113	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	dmeqi 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 )
7	4, 5, 6	3eqtri 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 )
11	8, 9, 10	3eqtri 2214	. . . 4 |- dom ( G o. `'inr ) = ran (inr |` dom G )
12	7, 11	uneq12i 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 )
14	12, 13	eqtri 2210	. 2 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( dom F ⊔ dom G )
15	2, 3, 14	3eqtri 2214	1 |- dom case ( F , G ) = ( dom F ⊔ dom G )

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