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Theorem djudm 6956
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 6954 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
21dmeqi 4708 . 2 dom (𝐹d 𝐺) = dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3 dmun 4714 . 2 dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))) = (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺)))
4 dmco 5015 . . . . 5 dom (𝐹(inl ↾ dom 𝐹)) = ((inl ↾ dom 𝐹) “ dom 𝐹)
5 imacnvcnv 4971 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹)
6 resima 4820 . . . . . 6 ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹)
7 df-ima 4520 . . . . . 6 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
86, 7eqtri 2136 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹)
94, 5, 83eqtri 2140 . . . 4 dom (𝐹(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹)
10 dmco 5015 . . . . 5 dom (𝐺(inr ↾ dom 𝐺)) = ((inr ↾ dom 𝐺) “ dom 𝐺)
11 imacnvcnv 4971 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺)
12 resima 4820 . . . . . 6 ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺)
13 df-ima 4520 . . . . . 6 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1412, 13eqtri 2136 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1510, 11, 143eqtri 2140 . . . 4 dom (𝐺(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺)
169, 15uneq12i 3196 . . 3 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
17 djuunr 6917 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1816, 17eqtri 2136 . 2 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺)
192, 3, 183eqtri 2140 1 dom (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1314  cun 3037  ccnv 4506  dom cdm 4507  ran crn 4508  cres 4509  cima 4510  ccom 4511  cdju 6888  inlcinl 6896  inrcinr 6897  d cdjud 6953
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899  df-djud 6954
This theorem is referenced by: (None)
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