Theorem djuunr 7183

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 7173	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		f1ofo 5541	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴))
3		forn 5513	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
4	1, 2, 3	mp2b 8	. . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5		djurf1or 7174	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
6		f1ofo 5541	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵))
7		forn 5513	. . . 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 3329	. 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
10		df-dju 7155	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
11	9, 10	eqtr4i 2230	1 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   = wceq 1373   ∪ cun 3168  ∅c0 3464  {csn 3638   × cxp 4681  ran crn 4684   ↾ cres 4685  –onto→wfo 5278  –1-1-onto→wf1o 5279  1_oc1o 6508   ⊔ cdju 7154  inlcinl 7162  inrcinr 7163

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1st 6239  df-2nd 6240  df-1o 6515  df-dju 7155  df-inl 7164  df-inr 7165

This theorem is referenced by:  djuun  7184  eldju  7185  casedm  7203  djudm  7222