Theorem casedm 7377

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 7375	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	dmeqi 4957	. 2 |- dom case ( F , G ) = dom ( ( F o. `'inl ) u. ( G o. `'inr ) )
3		dmun 4963	. 2 |- dom ( ( F o. `'inl ) u. ( G o. `'inr ) ) = ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) )
4		dmco 5271	. . . . 5 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
5		imacnvcnv 5227	. . . . 5 |- ( `' `'inl " dom F ) = (inl " dom F )
6		df-ima 4762	. . . . 5 |- (inl " dom F ) = ran (inl |` dom F )
7	4, 5, 6	3eqtri 2257	. . . 4 |- dom ( F o. `'inl ) = ran (inl |` dom F )
8		dmco 5271	. . . . 5 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
9		imacnvcnv 5227	. . . . 5 |- ( `' `'inr " dom G ) = (inr " dom G )
10		df-ima 4762	. . . . 5 |- (inr " dom G ) = ran (inr |` dom G )
11	8, 9, 10	3eqtri 2257	. . . 4 |- dom ( G o. `'inr ) = ran (inr |` dom G )
12	7, 11	uneq12i 3371	. . 3 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( ran (inl |` dom F ) u. ran (inr |` dom G ) )
13		djuunr 7357	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
14	12, 13	eqtri 2253	. 2 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( dom F ⊔ dom G )
15	2, 3, 14	3eqtri 2257	1 |- dom case ( F , G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1398   u. cun 3209   `'ccnv 4748   dom cdm 4749   ran crn 4750   |` cres 4751   "cima 4752   o. ccom 4753   ⊔ cdju 7328  inlcinl 7336  inrcinr 7337  casecdjucase 7374

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-1o 6647  df-dju 7329  df-inl 7338  df-inr 7339  df-case 7375

This theorem is referenced by:  casef  7379