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Theorem casedm 7084
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 7082 . . 3 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21dmeqi 4828 . 2 dom case(𝐹, 𝐺) = dom ((𝐹inl) ∪ (𝐺inr))
3 dmun 4834 . 2 dom ((𝐹inl) ∪ (𝐺inr)) = (dom (𝐹inl) ∪ dom (𝐺inr))
4 dmco 5137 . . . . 5 dom (𝐹inl) = (inl “ dom 𝐹)
5 imacnvcnv 5093 . . . . 5 (inl “ dom 𝐹) = (inl “ dom 𝐹)
6 df-ima 4639 . . . . 5 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
74, 5, 63eqtri 2202 . . . 4 dom (𝐹inl) = ran (inl ↾ dom 𝐹)
8 dmco 5137 . . . . 5 dom (𝐺inr) = (inr “ dom 𝐺)
9 imacnvcnv 5093 . . . . 5 (inr “ dom 𝐺) = (inr “ dom 𝐺)
10 df-ima 4639 . . . . 5 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
118, 9, 103eqtri 2202 . . . 4 dom (𝐺inr) = ran (inr ↾ dom 𝐺)
127, 11uneq12i 3287 . . 3 (dom (𝐹inl) ∪ dom (𝐺inr)) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
13 djuunr 7064 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1412, 13eqtri 2198 . 2 (dom (𝐹inl) ∪ dom (𝐺inr)) = (dom 𝐹 ⊔ dom 𝐺)
152, 3, 143eqtri 2202 1 dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cun 3127  ccnv 4625  dom cdm 4626  ran crn 4627  cres 4628  cima 4629  ccom 4630  cdju 7035  inlcinl 7043  inrcinr 7044  casecdjucase 7081
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  casef  7086
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