Theorem djuun 7413

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 4789	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4789	. . 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 7412	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
5	3, 4	eqtri 2259	1 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

This proof depends on syntax axioms:   = wceq 1402   u. cun 3218   ran crn 4777   |` cres 4778   "cima 4779   ⊔ cdju 7383  inlcinl 7391  inrcinr 7392

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394

This theorem is used by:  endjusym  7442  ctssdccl  7457  dju1p1e2  7555  endjudisj  7572  djuen  7573