Theorem djuun 7044

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 4624	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4624	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	uneq12i 3279	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( ran (inl |` A ) u. ran (inr |` B ) )
4		djuunr 7043	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
5	3, 4	eqtri 2191	1 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1348   u. cun 3119   ran crn 4612   |` cres 4613   "cima 4614   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  endjusym  7073  ctssdccl  7088  dju1p1e2  7174  endjudisj  7187  djuen  7188