Theorem djulclr 7204

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 5647	. 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶))
2		elex 2811	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
3		0ex 4210	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 3697	. . . . 5 ⊢ ∅ ∈ {∅}
5		opelxpi 4748	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7		opeq2 3857	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
8		df-inl 7202	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
9	7, 8	fvmptg 5703	. . . 4 ⊢ ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10	2, 6, 9	syl2anc 411	. . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
11		elun1 3371	. . . . 5 ⊢ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12	6, 11	syl 14	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13		df-dju 7193	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
14	12, 13	eleqtrrdi 2323	. . 3 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
15	10, 14	eqeltrd 2306	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
16	1, 15	eqeltrd 2306	1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195  ∅c0 3491  {csn 3666  ⟨cop 3669   × cxp 4714   ↾ cres 4718  ‘cfv 5314  1_oc1o 6545   ⊔ cdju 7192  inlcinl 7200

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-res 4728  df-iota 5274  df-fun 5316  df-fv 5322  df-dju 7193  df-inl 7202

This theorem is referenced by:  inlresf1  7216