Theorem casedm 6971

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 6969	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	dmeqi 4740	. 2 ⊢ dom case(𝐹, 𝐺) = dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
3		dmun 4746	. 2 ⊢ dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)) = (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr))
4		dmco 5047	. . . . 5 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
5		imacnvcnv 5003	. . . . 5 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
6		df-ima 4552	. . . . 5 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
7	4, 5, 6	3eqtri 2164	. . . 4 ⊢ dom (𝐹 ∘ ◡inl) = ran (inl ↾ dom 𝐹)
8		dmco 5047	. . . . 5 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
9		imacnvcnv 5003	. . . . 5 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
10		df-ima 4552	. . . . 5 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
11	8, 9, 10	3eqtri 2164	. . . 4 ⊢ dom (𝐺 ∘ ◡inr) = ran (inr ↾ dom 𝐺)
12	7, 11	uneq12i 3228	. . 3 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13		djuunr 6951	. . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
14	12, 13	eqtri 2160	. 2 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (dom 𝐹 ⊔ dom 𝐺)
15	2, 3, 14	3eqtri 2164	1 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Syntax hints:   = wceq 1331   ∪ cun 3069  ^◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   “ cima 4542   ∘ ccom 4543   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931  casecdjucase 6968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-case 6969

This theorem is referenced by:  casef  6973