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Theorem djuun 6918
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) ∪ (inr “ 𝐵)) = (𝐴𝐵)

Proof of Theorem djuun
StepHypRef Expression
1 df-ima 4520 . . 3 (inl “ 𝐴) = ran (inl ↾ 𝐴)
2 df-ima 4520 . . 3 (inr “ 𝐵) = ran (inr ↾ 𝐵)
31, 2uneq12i 3196 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
4 djuunr 6917 . 2 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
53, 4eqtri 2136 1 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1314  cun 3037  ran crn 4508  cres 4509  cima 4510  cdju 6888  inlcinl 6896  inrcinr 6897
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
This theorem is referenced by:  endjusym  6947  ctssdccl  6962  dju1p1e2  7017  endjudisj  7030  djuen  7031
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