Theorem casedm 7276

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 7274	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	dmeqi 4930	. 2 ⊢ dom case(𝐹, 𝐺) = dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
3		dmun 4936	. 2 ⊢ dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)) = (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr))
4		dmco 5243	. . . . 5 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
5		imacnvcnv 5199	. . . . 5 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
6		df-ima 4736	. . . . 5 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
7	4, 5, 6	3eqtri 2254	. . . 4 ⊢ dom (𝐹 ∘ ◡inl) = ran (inl ↾ dom 𝐹)
8		dmco 5243	. . . . 5 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
9		imacnvcnv 5199	. . . . 5 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
10		df-ima 4736	. . . . 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 3357	. . 3 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13		djuunr 7256	. . 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 3196  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   ↾ cres 4725   “ cima 4726   ∘ ccom 4727   ⊔ cdju 7227  inlcinl 7235  inrcinr 7236  casecdjucase 7273

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-1o 6577  df-dju 7228  df-inl 7237  df-inr 7238  df-case 7274

This theorem is referenced by:  casef  7278