Theorem casedm 7051

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 7049	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	dmeqi 4805	. 2 |- dom case ( F , G ) = dom ( ( F o. `'inl ) u. ( G o. `'inr ) )
3		dmun 4811	. 2 |- dom ( ( F o. `'inl ) u. ( G o. `'inr ) ) = ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) )
4		dmco 5112	. . . . 5 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
5		imacnvcnv 5068	. . . . 5 |- ( `' `'inl " dom F ) = (inl " dom F )
6		df-ima 4617	. . . . 5 |- (inl " dom F ) = ran (inl |` dom F )
7	4, 5, 6	3eqtri 2190	. . . 4 |- dom ( F o. `'inl ) = ran (inl |` dom F )
8		dmco 5112	. . . . 5 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
9		imacnvcnv 5068	. . . . 5 |- ( `' `'inr " dom G ) = (inr " dom G )
10		df-ima 4617	. . . . 5 |- (inr " dom G ) = ran (inr |` dom G )
11	8, 9, 10	3eqtri 2190	. . . 4 |- dom ( G o. `'inr ) = ran (inr |` dom G )
12	7, 11	uneq12i 3274	. . 3 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( ran (inl |` dom F ) u. ran (inr |` dom G ) )
13		djuunr 7031	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
14	12, 13	eqtri 2186	. 2 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( dom F ⊔ dom G )
15	2, 3, 14	3eqtri 2190	1 |- dom case ( F , G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1343   u. cun 3114   `'ccnv 4603   dom cdm 4604   ran crn 4605   |` cres 4606   "cima 4607   o. ccom 4608   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011  casecdjucase 7048

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef  7053