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Theorem djuun 7059
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 4635 . . 3 (inl “ 𝐴) = ran (inl ↾ 𝐴)
2 df-ima 4635 . . 3 (inr “ 𝐵) = ran (inr ↾ 𝐵)
31, 2uneq12i 3287 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
4 djuunr 7058 . 2 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
53, 4eqtri 2198 1 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cun 3127  ran crn 4623  cres 4624  cima 4625  cdju 7029  inlcinl 7037  inrcinr 7038
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-1st 6134  df-2nd 6135  df-1o 6410  df-dju 7030  df-inl 7039  df-inr 7040
This theorem is referenced by:  endjusym  7088  ctssdccl  7103  dju1p1e2  7189  endjudisj  7202  djuen  7203
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