Theorem djuun 7400

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 4785	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4785	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	uneq12i 3381	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( ran (inl |` A ) u. ran (inr |` B ) )
4		djuunr 7399	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
5	3, 4	eqtri 2259	1 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1402   u. cun 3218   ran crn 4773   |` cres 4774   "cima 4775   ⊔ cdju 7370  inlcinl 7378  inrcinr 7379

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6367  df-2nd 6368  df-1o 6680  df-dju 7371  df-inl 7380  df-inr 7381

This theorem is referenced by:  endjusym  7429  ctssdccl  7444  dju1p1e2  7542  endjudisj  7559  djuen  7560