Theorem casedm 7087

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 7085	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	dmeqi 4830	. 2 |- dom case ( F , G ) = dom ( ( F o. `'inl ) u. ( G o. `'inr ) )
3		dmun 4836	. 2 |- dom ( ( F o. `'inl ) u. ( G o. `'inr ) ) = ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) )
4		dmco 5139	. . . . 5 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
5		imacnvcnv 5095	. . . . 5 |- ( `' `'inl " dom F ) = (inl " dom F )
6		df-ima 4641	. . . . 5 |- (inl " dom F ) = ran (inl |` dom F )
7	4, 5, 6	3eqtri 2202	. . . 4 |- dom ( F o. `'inl ) = ran (inl |` dom F )
8		dmco 5139	. . . . 5 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
9		imacnvcnv 5095	. . . . 5 |- ( `' `'inr " dom G ) = (inr " dom G )
10		df-ima 4641	. . . . 5 |- (inr " dom G ) = ran (inr |` dom G )
11	8, 9, 10	3eqtri 2202	. . . 4 |- dom ( G o. `'inr ) = ran (inr |` dom G )
12	7, 11	uneq12i 3289	. . 3 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( ran (inl |` dom F ) u. ran (inr |` dom G ) )
13		djuunr 7067	. . 3 |- ( ran (inl |` dom F ) u. ran (inr |` dom G ) ) = ( dom F ⊔ dom G )
14	12, 13	eqtri 2198	. 2 |- ( dom ( F o. `'inl ) u. dom ( G o. `'inr ) ) = ( dom F ⊔ dom G )
15	2, 3, 14	3eqtri 2202	1 |- dom case ( F , G ) = ( dom F ⊔ dom G )

Syntax hints:   = wceq 1353   u. cun 3129   `'ccnv 4627   dom cdm 4628   ran crn 4629   |` cres 4630   "cima 4631   o. ccom 4632   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047  casecdjucase 7084

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049  df-case 7085

This theorem is referenced by:  casef  7089