Theorem djulclr 7383

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

Assertion

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

Proof of Theorem djulclr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 5717	. 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶))
2		elex 2833	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
3		0ex 4258	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 3739	. . . . 5 ⊢ ∅ ∈ {∅}
5		opelxpi 4804	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6	4, 5	mpan 428	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7		opeq2 3903	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
8		df-inl 7381	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
9	7, 8	fvmptg 5778	. . . 4 ⊢ ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10	2, 6, 9	syl2anc 415	. . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
11		elun1 3396	. . . . 5 ⊢ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12	6, 11	syl 14	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13		df-dju 7372	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
14	12, 13	eleqtrrdi 2332	. . 3 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
15	10, 14	eqeltrd 2315	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
16	1, 15	eqeltrd 2315	1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218  ∅c0 3520  {csn 3708  ⟨cop 3711   × cxp 4770   ↾ cres 4774  ‘cfv 5375  1_oc1o 6674   ⊔ cdju 7371  inlcinl 7379

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-res 4784  df-iota 5335  df-fun 5377  df-fv 5383  df-dju 7372  df-inl 7381

This theorem is referenced by:  inlresf1  7395