Theorem casedm 7249

Description: The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): ⊢ ran case(𝐹, 𝐺) = (ran 𝐹 ∪ 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 7247	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	dmeqi 4923	. 2 ⊢ dom case(𝐹, 𝐺) = dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
3		dmun 4929	. 2 ⊢ dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)) = (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr))
4		dmco 5236	. . . . 5 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
5		imacnvcnv 5192	. . . . 5 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
6		df-ima 4731	. . . . 5 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
7	4, 5, 6	3eqtri 2254	. . . 4 ⊢ dom (𝐹 ∘ ◡inl) = ran (inl ↾ dom 𝐹)
8		dmco 5236	. . . . 5 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
9		imacnvcnv 5192	. . . . 5 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
10		df-ima 4731	. . . . 5 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
11	8, 9, 10	3eqtri 2254	. . . 4 ⊢ dom (𝐺 ∘ ◡inr) = ran (inr ↾ dom 𝐺)
12	7, 11	uneq12i 3356	. . 3 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13		djuunr 7229	. . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
14	12, 13	eqtri 2250	. 2 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (dom 𝐹 ⊔ dom 𝐺)
15	2, 3, 14	3eqtri 2254	1 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Syntax hints:   = wceq 1395   ∪ cun 3195  ^◡ccnv 4717  dom cdm 4718  ran crn 4719   ↾ cres 4720   “ cima 4721   ∘ ccom 4722   ⊔ cdju 7200  inlcinl 7208  inrcinr 7209  casecdjucase 7246

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-1o 6560  df-dju 7201  df-inl 7210  df-inr 7211  df-case 7247

This theorem is referenced by:  casef  7251