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Theorem djudm 6764
Description: The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
djudm dom (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Proof of Theorem djudm
StepHypRef Expression
1 df-djud 6762 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
21dmeqi 4625 . 2 dom (𝐹d 𝐺) = dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3 dmun 4631 . 2 dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))) = (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺)))
4 dmco 4926 . . . . 5 dom (𝐹(inl ↾ dom 𝐹)) = ((inl ↾ dom 𝐹) “ dom 𝐹)
5 imacnvcnv 4882 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹)
6 resima 4732 . . . . . 6 ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹)
7 df-ima 4441 . . . . . 6 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
86, 7eqtri 2108 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹)
94, 5, 83eqtri 2112 . . . 4 dom (𝐹(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹)
10 dmco 4926 . . . . 5 dom (𝐺(inr ↾ dom 𝐺)) = ((inr ↾ dom 𝐺) “ dom 𝐺)
11 imacnvcnv 4882 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺)
12 resima 4732 . . . . . 6 ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺)
13 df-ima 4441 . . . . . 6 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1412, 13eqtri 2108 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1510, 11, 143eqtri 2112 . . . 4 dom (𝐺(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺)
169, 15uneq12i 3150 . . 3 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
17 djuunr 6737 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1816, 17eqtri 2108 . 2 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺)
192, 3, 183eqtri 2112 1 dom (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2995  ccnv 4427  dom cdm 4428  ran crn 4429  cres 4430  cima 4431  ccom 4432  cdju 6709  inlcinl 6716  inrcinr 6717  d cdjud 6761
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 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
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 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719  df-djud 6762
This theorem is referenced by: (None)
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