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Theorem djuunr 7127
Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djuunr (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)

Proof of Theorem djuunr
StepHypRef Expression
1 djulf1or 7117 . . . 4 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 f1ofo 5508 . . . 4 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴onto→({∅} × 𝐴))
3 forn 5480 . . . 4 ((inl ↾ 𝐴):𝐴onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
41, 2, 3mp2b 8 . . 3 ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5 djurf1or 7118 . . . 4 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
6 f1ofo 5508 . . . 4 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵onto→({1o} × 𝐵))
7 forn 5480 . . . 4 ((inr ↾ 𝐵):𝐵onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵))
85, 6, 7mp2b 8 . . 3 ran (inr ↾ 𝐵) = ({1o} × 𝐵)
94, 8uneq12i 3312 . 2 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
10 df-dju 7099 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
119, 10eqtr4i 2217 1 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  cun 3152  c0 3447  {csn 3619   × cxp 4658  ran crn 4661  cres 4662  ontowfo 5253  1-1-ontowf1o 5254  1oc1o 6464  cdju 7098  inlcinl 7106  inrcinr 7107
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109
This theorem is referenced by:  djuun  7128  eldju  7129  casedm  7147  djudm  7166
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