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Theorem casedm 7020
 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
StepHypRef Expression
1 df-case 7018 . . 3 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21dmeqi 4784 . 2 dom case(𝐹, 𝐺) = dom ((𝐹inl) ∪ (𝐺inr))
3 dmun 4790 . 2 dom ((𝐹inl) ∪ (𝐺inr)) = (dom (𝐹inl) ∪ dom (𝐺inr))
4 dmco 5091 . . . . 5 dom (𝐹inl) = (inl “ dom 𝐹)
5 imacnvcnv 5047 . . . . 5 (inl “ dom 𝐹) = (inl “ dom 𝐹)
6 df-ima 4596 . . . . 5 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
74, 5, 63eqtri 2182 . . . 4 dom (𝐹inl) = ran (inl ↾ dom 𝐹)
8 dmco 5091 . . . . 5 dom (𝐺inr) = (inr “ dom 𝐺)
9 imacnvcnv 5047 . . . . 5 (inr “ dom 𝐺) = (inr “ dom 𝐺)
10 df-ima 4596 . . . . 5 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
118, 9, 103eqtri 2182 . . . 4 dom (𝐺inr) = ran (inr ↾ dom 𝐺)
127, 11uneq12i 3259 . . 3 (dom (𝐹inl) ∪ dom (𝐺inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13 djuunr 7000 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1412, 13eqtri 2178 . 2 (dom (𝐹inl) ∪ dom (𝐺inr)) = (dom 𝐹 ⊔ dom 𝐺)
152, 3, 143eqtri 2182 1 dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
 Colors of variables: wff set class Syntax hints:   = wceq 1335   ∪ cun 3100  ◡ccnv 4582  dom cdm 4583  ran crn 4584   ↾ cres 4585   “ cima 4586   ∘ ccom 4587   ⊔ cdju 6971  inlcinl 6979  inrcinr 6980  casecdjucase 7017 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-dju 6972  df-inl 6981  df-inr 6982  df-case 7018 This theorem is referenced by:  casef  7022
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