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Theorem casedm 6767
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 6765 . . 3 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21dmeqi 4633 . 2 dom case(𝐹, 𝐺) = dom ((𝐹inl) ∪ (𝐺inr))
3 dmun 4639 . 2 dom ((𝐹inl) ∪ (𝐺inr)) = (dom (𝐹inl) ∪ dom (𝐺inr))
4 dmco 4934 . . . . 5 dom (𝐹inl) = (inl “ dom 𝐹)
5 imacnvcnv 4890 . . . . 5 (inl “ dom 𝐹) = (inl “ dom 𝐹)
6 df-ima 4449 . . . . 5 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
74, 5, 63eqtri 2112 . . . 4 dom (𝐹inl) = ran (inl ↾ dom 𝐹)
8 dmco 4934 . . . . 5 dom (𝐺inr) = (inr “ dom 𝐺)
9 imacnvcnv 4890 . . . . 5 (inr “ dom 𝐺) = (inr “ dom 𝐺)
10 df-ima 4449 . . . . 5 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
118, 9, 103eqtri 2112 . . . 4 dom (𝐺inr) = ran (inr ↾ dom 𝐺)
127, 11uneq12i 3152 . . 3 (dom (𝐹inl) ∪ dom (𝐺inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13 djuunr 6748 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1412, 13eqtri 2108 . 2 (dom (𝐹inl) ∪ dom (𝐺inr)) = (dom 𝐹 ⊔ dom 𝐺)
152, 3, 143eqtri 2112 1 dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2997  ccnv 4435  dom cdm 4436  ran crn 4437  cres 4438  cima 4439  ccom 4440  cdju 6720  inlcinl 6727  inrcinr 6728  casecdjucase 6764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-1st 5903  df-2nd 5904  df-1o 6173  df-dju 6721  df-inl 6729  df-inr 6730  df-case 6765
This theorem is referenced by:  casef  6769
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