Theorem djuunr 7370

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, 6-Jul-2022.)

Assertion

Ref	Expression
djuunr	⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)

Proof of Theorem djuunr

Step	Hyp	Ref	Expression
1		djulf1or 7360	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		f1ofo 5626	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴))
3		forn 5598	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
4	1, 2, 3	mp2b 8	. . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5		djurf1or 7361	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
6		f1ofo 5626	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵))
7		forn 5598	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵))
8	5, 6, 7	mp2b 8	. . 3 ⊢ ran (inr ↾ 𝐵) = ({1o} × 𝐵)
9	4, 8	uneq12i 3375	. 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
10		df-dju 7342	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
11	9, 10	eqtr4i 2258	1 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   = wceq 1398   ∪ cun 3212  ∅c0 3512  {csn 3694   × cxp 4752  ran crn 4755   ↾ cres 4756  –onto→wfo 5355  –1-1-onto→wf1o 5356  1_oc1o 6653   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  djuun  7371  eldju  7372  casedm  7390  djudm  7409