ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  djudm GIF version

Theorem djudm 7051
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 7049 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
21dmeqi 4789 . 2 dom (𝐹d 𝐺) = dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3 dmun 4795 . 2 dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))) = (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺)))
4 dmco 5096 . . . . 5 dom (𝐹(inl ↾ dom 𝐹)) = ((inl ↾ dom 𝐹) “ dom 𝐹)
5 imacnvcnv 5052 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹)
6 resima 4901 . . . . . 6 ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹)
7 df-ima 4601 . . . . . 6 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
86, 7eqtri 2178 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹)
94, 5, 83eqtri 2182 . . . 4 dom (𝐹(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹)
10 dmco 5096 . . . . 5 dom (𝐺(inr ↾ dom 𝐺)) = ((inr ↾ dom 𝐺) “ dom 𝐺)
11 imacnvcnv 5052 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺)
12 resima 4901 . . . . . 6 ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺)
13 df-ima 4601 . . . . . 6 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1412, 13eqtri 2178 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1510, 11, 143eqtri 2182 . . . 4 dom (𝐺(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺)
169, 15uneq12i 3260 . . 3 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
17 djuunr 7012 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1816, 17eqtri 2178 . 2 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺)
192, 3, 183eqtri 2182 1 dom (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1335  cun 3100  ccnv 4587  dom cdm 4588  ran crn 4589  cres 4590  cima 4591  ccom 4592  cdju 6983  inlcinl 6991  inrcinr 6992  d cdjud 7048
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 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395
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 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-1st 6090  df-2nd 6091  df-1o 6365  df-dju 6984  df-inl 6993  df-inr 6994  df-djud 7049
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator