Theorem casedm 7390

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 7388	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	dmeqi 4962	. 2 |- dom case ( F , G ) = dom ( ( F o. `'inl ) u. ( G o. `'inr ) )
3		dmun 4968	. 2 |- dom ( ( F o. `'inl ) u. ( G o. `'inr ) ) = ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) )
4		dmco 5276	. . . . 5 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
5		imacnvcnv 5232	. . . . 5 |- ( `' `'inl " dom F ) = (inl " dom F )
6		df-ima 4767	. . . . 5 |- (inl " dom F ) = ran (inl |` dom F )
7	4, 5, 6	3eqtri 2259	. . . 4 |- dom ( F o. `'inl ) = ran (inl |` dom F )
8		dmco 5276	. . . . 5 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
9		imacnvcnv 5232	. . . . 5 |- ( `' `'inr " dom G ) = (inr " dom G )
10		df-ima 4767	. . . . 5 |- (inr " dom G ) = ran (inr |` dom G )
11	8, 9, 10	3eqtri 2259	. . . 4 |- dom ( G o. `'inr ) = ran (inr |` dom G )
12	7, 11	uneq12i 3375	. . 3 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( ran (inl |` dom F ) u. ran (inr |` dom G ) )
13		djuunr 7370	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
14	12, 13	eqtri 2255	. 2 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( dom F ⊔ dom G )
15	2, 3, 14	3eqtri 2259	1 |- dom case ( F , G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1398   u. cun 3212   `'ccnv 4753   dom cdm 4754   ran crn 4755   |` cres 4756   "cima 4757   o. ccom 4758   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350  casecdjucase 7387

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352  df-case 7388

This theorem is referenced by:  casef  7392