Theorem djulcl 9948

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

Assertion

Ref	Expression
djulcl	⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Proof of Theorem djulcl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 9940	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 4879	. . 3 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
3		elex 3499	. . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
4		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
5	4	snid 4667	. . . 4 ⊢ ∅ ∈ {∅}
6		opelxpi 5726	. . . 4 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7	5, 6	mpan 690	. . 3 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
8	1, 2, 3, 7	fvmptd3 7039	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
9		elun1 4192	. . . 4 ⊢ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
10	7, 9	syl 17	. . 3 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
11		df-dju 9939	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
12	10, 11	eleqtrrdi 2850	. 2 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
13	8, 12	eqeltrd 2839	1 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961  ∅c0 4339  {csn 4631  ⟨cop 4637   × cxp 5687  ‘cfv 6563  1_oc1o 8498   ⊔ cdju 9936  inlcinl 9937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-dju 9939  df-inl 9940

This theorem is referenced by:  inlresf  9952  updjudhcoinlf  9970