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

Theorem djudm 7106
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 7104 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
21dmeqi 4830 . 2 dom (𝐹d 𝐺) = dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3 dmun 4836 . 2 dom ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))) = (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺)))
4 dmco 5139 . . . . 5 dom (𝐹(inl ↾ dom 𝐹)) = ((inl ↾ dom 𝐹) “ dom 𝐹)
5 imacnvcnv 5095 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹)
6 resima 4942 . . . . . 6 ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹)
7 df-ima 4641 . . . . . 6 (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹)
86, 7eqtri 2198 . . . . 5 ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹)
94, 5, 83eqtri 2202 . . . 4 dom (𝐹(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹)
10 dmco 5139 . . . . 5 dom (𝐺(inr ↾ dom 𝐺)) = ((inr ↾ dom 𝐺) “ dom 𝐺)
11 imacnvcnv 5095 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺)
12 resima 4942 . . . . . 6 ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺)
13 df-ima 4641 . . . . . 6 (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1412, 13eqtri 2198 . . . . 5 ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺)
1510, 11, 143eqtri 2202 . . . 4 dom (𝐺(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺)
169, 15uneq12i 3289 . . 3 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺))
17 djuunr 7067 . . 3 (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺)
1816, 17eqtri 2198 . 2 (dom (𝐹(inl ↾ dom 𝐹)) ∪ dom (𝐺(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺)
192, 3, 183eqtri 2202 1 dom (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cun 3129  ccnv 4627  dom cdm 4628  ran crn 4629  cres 4630  cima 4631  ccom 4632  cdju 7038  inlcinl 7046  inrcinr 7047  d cdjud 7103
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049  df-djud 7104
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator