Theorem casedm 7085

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 7083	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	dmeqi 4829	. 2 ⊢ dom case(𝐹, 𝐺) = dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
3		dmun 4835	. 2 ⊢ dom ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)) = (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr))
4		dmco 5138	. . . . 5 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
5		imacnvcnv 5094	. . . . 5 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
6		df-ima 4640	. . . . 5 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
7	4, 5, 6	3eqtri 2202	. . . 4 ⊢ dom (𝐹 ∘ ◡inl) = ran (inl ↾ dom 𝐹)
8		dmco 5138	. . . . 5 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
9		imacnvcnv 5094	. . . . 5 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
10		df-ima 4640	. . . . 5 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
11	8, 9, 10	3eqtri 2202	. . . 4 ⊢ dom (𝐺 ∘ ◡inr) = ran (inr ↾ dom 𝐺)
12	7, 11	uneq12i 3288	. . 3 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13		djuunr 7065	. . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
14	12, 13	eqtri 2198	. 2 ⊢ (dom (𝐹 ∘ ◡inl) ∪ dom (𝐺 ∘ ◡inr)) = (dom 𝐹 ⊔ dom 𝐺)
15	2, 3, 14	3eqtri 2202	1 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Syntax hints:   = wceq 1353   ∪ cun 3128  ^◡ccnv 4626  dom cdm 4627  ran crn 4628   ↾ cres 4629   “ cima 4630   ∘ ccom 4631   ⊔ cdju 7036  inlcinl 7044  inrcinr 7045  casecdjucase 7082

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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434

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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047  df-case 7083

This theorem is referenced by:  casef  7087