Theorem djuun 7011

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 4599	. . 3 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴)
2		df-ima 4599	. . 3 ⊢ (inr “ 𝐵) = ran (inr ↾ 𝐵)
3	1, 2	uneq12i 3259	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
4		djuunr 7010	. 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
5	3, 4	eqtri 2178	1 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   = wceq 1335   ∪ cun 3100  ran crn 4587   ↾ cres 4588   “ cima 4589   ⊔ cdju 6981  inlcinl 6989  inrcinr 6990

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-1o 6363  df-dju 6982  df-inl 6991  df-inr 6992

This theorem is referenced by:  endjusym  7040  ctssdccl  7055  dju1p1e2  7132  endjudisj  7145  djuen  7146