Theorem djulclr 7339

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 5693	. 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶))
2		elex 2824	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
3		0ex 4236	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 3719	. . . . 5 ⊢ ∅ ∈ {∅}
5		opelxpi 4780	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7		opeq2 3883	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
8		df-inl 7337	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
9	7, 8	fvmptg 5752	. . . 4 ⊢ ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10	2, 6, 9	syl2anc 411	. . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
11		elun1 3385	. . . . 5 ⊢ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12	6, 11	syl 14	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13		df-dju 7328	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
14	12, 13	eleqtrrdi 2326	. . 3 ⊢ (𝐶 ∈ 𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴 ⊔ 𝐵))
15	10, 14	eqeltrd 2309	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
16	1, 15	eqeltrd 2309	1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∪ cun 3208  ∅c0 3507  {csn 3688  ⟨cop 3691   × cxp 4746   ↾ cres 4750  ‘cfv 5351  1_oc1o 6639   ⊔ cdju 7327  inlcinl 7335

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-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321

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 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-res 4760  df-iota 5311  df-fun 5353  df-fv 5359  df-dju 7328  df-inl 7337

This theorem is referenced by:  inlresf1  7351