Theorem djurcl 10019

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 3475	. . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V)
2		1oex 8514	. . . . 5 ⊢ 1o ∈ V
3	2	snid 4623	. . . 4 ⊢ 1o ∈ {1o}
4		opelxpi 5692	. . . 4 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
5	3, 4	mpan 703	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6		opeq2 4844	. . . 4 ⊢ (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7		df-inr 10011	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
8	6, 7	fvmptg 7014	. . 3 ⊢ ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
9	1, 5, 8	syl2anc 596	. 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10		elun2 4184	. . . 4 ⊢ (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
11	5, 10	syl 18	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12		df-dju 10009	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
13	11, 12	eleqtrrdi 2875	. 2 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
14	9, 13	eqeltrd 2864	1 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838   × cxp 5653  ‘cfv 6558  1_oc1o 8497   ⊔ cdju 10006  inrcinr 10008

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-suc 6386  df-iota 6512  df-fun 6560  df-fv 6566  df-1o 8504  df-dju 10009  df-inr 10011

This theorem is used by:  inrresf  10024  updjudhcoinrg  10041