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Theorem djuun 6920
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, 9-Jul-2023.)
Assertion
Ref Expression
djuun  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )

Proof of Theorem djuun
StepHypRef Expression
1 df-ima 4522 . . 3  |-  (inl " A )  =  ran  (inl  |`  A )
2 df-ima 4522 . . 3  |-  (inr " B )  =  ran  (inr  |`  B )
31, 2uneq12i 3198 . 2  |-  ( (inl " A )  u.  (inr " B ) )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
4 djuunr 6919 . 2  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
53, 4eqtri 2138 1  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    u. cun 3039   ran crn 4510    |` cres 4511   "cima 4512   ⊔ cdju 6890  inlcinl 6898  inrcinr 6899
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  endjusym  6949  ctssdccl  6964  dju1p1e2  7021  endjudisj  7034  djuen  7035
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