Theorem djuun 6902

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

Step	Hyp	Ref	Expression
1		df-ima 4510	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4510	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	uneq12i 3192	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( ran (inl |` A ) u. ran (inr |` B ) )
4		djuunr 6901	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
5	3, 4	eqtri 2133	1 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1312   u. cun 3033   ran crn 4498   |` cres 4499   "cima 4500   ⊔ cdju 6872  inlcinl 6880  inrcinr 6881

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-dju 6873  df-inl 6882  df-inr 6883

This theorem is referenced by:  endjusym  6931  ctssdccl  6946  dju1p1e2  6998  endjudisj  7011  djuen  7012