Theorem djurcl 9669

Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djurcl	⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Proof of Theorem djurcl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V)
2		1oex 8307	. . . . 5 ⊢ 1o ∈ V
3	2	snid 4597	. . . 4 ⊢ 1o ∈ {1o}
4		opelxpi 5626	. . . 4 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
5	3, 4	mpan 687	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6		opeq2 4805	. . . 4 ⊢ (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7		df-inr 9661	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
8	6, 7	fvmptg 6873	. . 3 ⊢ ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
9	1, 5, 8	syl2anc 584	. 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10		elun2 4111	. . . 4 ⊢ (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
11	5, 10	syl 17	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12		df-dju 9659	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
13	11, 12	eleqtrrdi 2850	. 2 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
14	9, 13	eqeltrd 2839	1 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885  ∅c0 4256  {csn 4561  ⟨cop 4567   × cxp 5587  ‘cfv 6433  1_oc1o 8290   ⊔ cdju 9656  inrcinr 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-1o 8297  df-dju 9659  df-inr 9661

This theorem is referenced by:  inrresf  9674  updjudhcoinrg  9691