Theorem djurclr 7166

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 5612	. 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶))
2		elex 2785	. . . 4 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V)
3		1oex 6522	. . . . . 6 ⊢ 1o ∈ V
4	3	snid 3668	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 4714	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
7		opeq2 3825	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
8		df-inr 7164	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
9	7, 8	fvmptg 5667	. . . 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 3345	. . . . 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 7154	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
14	12, 13	eleqtrrdi 2300	. . 3 ⊢ (𝐶 ∈ 𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
15	10, 14	eqeltrd 2283	. 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
16	1, 15	eqeltrd 2283	1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∪ cun 3168  ∅c0 3464  {csn 3637  ⟨cop 3640   × cxp 4680   ↾ cres 4684  ‘cfv 5279  1_oc1o 6507   ⊔ cdju 7153  inrcinr 7162

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 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487

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 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-res 4694  df-iota 5240  df-fun 5281  df-fv 5287  df-1o 6514  df-dju 7154  df-inr 7164

This theorem is referenced by:  inrresf1  7178