Theorem djurclr 7341

Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)

Assertion

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

Proof of Theorem djurclr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 5694	. 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶))
2		elex 2825	. . . 4 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V)
3		1oex 6655	. . . . . 6 ⊢ 1o ∈ V
4	3	snid 3720	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 4781	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
7		opeq2 3884	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
8		df-inr 7339	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
9	7, 8	fvmptg 5753	. . . 4 ⊢ ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10	2, 6, 9	syl2anc 411	. . 3 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
11		elun2 3387	. . . . 5 ⊢ (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12	6, 11	syl 14	. . . 4 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13		df-dju 7329	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
14	12, 13	eleqtrrdi 2326	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
15	10, 14	eqeltrd 2309	. 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
16	1, 15	eqeltrd 2309	1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813   ∪ cun 3209  ∅c0 3508  {csn 3689  ⟨cop 3692   × cxp 4747   ↾ cres 4751  ‘cfv 5352  1_oc1o 6640   ⊔ cdju 7328  inrcinr 7337

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-1o 6647  df-dju 7329  df-inr 7339

This theorem is referenced by:  inrresf1  7353