Theorem djuun 7255

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)) = (𝐴 ⊔ 𝐵)

Proof of Theorem djuun

Step	Hyp	Ref	Expression
1		df-ima 4734	. . 3 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴)
2		df-ima 4734	. . 3 ⊢ (inr “ 𝐵) = ran (inr ↾ 𝐵)
3	1, 2	uneq12i 3357	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
4		djuunr 7254	. 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
5	3, 4	eqtri 2250	1 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   = wceq 1395   ∪ cun 3196  ran crn 4722   ↾ cres 4723   “ cima 4724   ⊔ cdju 7225  inlcinl 7233  inrcinr 7234

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-suc 4464  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-1st 6296  df-2nd 6297  df-1o 6575  df-dju 7226  df-inl 7235  df-inr 7236

This theorem is referenced by:  endjusym  7284  ctssdccl  7299  dju1p1e2  7396  endjudisj  7413  djuen  7414