Theorem djurcl 9864

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 3468	. . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V)
2		1oex 8444	. . . . 5 ⊢ 1o ∈ V
3	2	snid 4626	. . . 4 ⊢ 1o ∈ {1o}
4		opelxpi 5675	. . . 4 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
5	3, 4	mpan 690	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6		opeq2 4838	. . . 4 ⊢ (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7		df-inr 9856	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
8	6, 7	fvmptg 6966	. . 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 4146	. . . 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 9854	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
13	11, 12	eleqtrrdi 2839	. 2 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
14	9, 13	eqeltrd 2828	1 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912  ∅c0 4296  {csn 4589  ⟨cop 4595   × cxp 5636  ‘cfv 6511  1_oc1o 8427   ⊔ cdju 9851  inrcinr 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-suc 6338  df-iota 6464  df-fun 6513  df-fv 6519  df-1o 8434  df-dju 9854  df-inr 9856

This theorem is referenced by:  inrresf  9869  updjudhcoinrg  9886