Theorem djuunr 7229

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 7219	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		f1ofo 5578	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴))
3		forn 5550	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
4	1, 2, 3	mp2b 8	. . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5		djurf1or 7220	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
6		f1ofo 5578	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵))
7		forn 5550	. . . 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 3356	. 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
10		df-dju 7201	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
11	9, 10	eqtr4i 2253	1 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   = wceq 1395   ∪ cun 3195  ∅c0 3491  {csn 3666   × cxp 4716  ran crn 4719   ↾ cres 4720  –onto→wfo 5315  –1-1-onto→wf1o 5316  1_oc1o 6553   ⊔ cdju 7200  inlcinl 7208  inrcinr 7209

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-1o 6560  df-dju 7201  df-inl 7210  df-inr 7211

This theorem is referenced by:  djuun  7230  eldju  7231  casedm  7249  djudm  7268