Theorem inlresf 9838

Description: The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
inlresf	⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)

Proof of Theorem inlresf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djulf1o 9836	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1ofun 6782	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
3		ffvresb 7078	. . 3 ⊢ (Fun inl → ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))))
4	1, 2, 3	mp2b 10	. 2 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)))
5		elex 3450	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V)
6		opex 5416	. . . . 5 ⊢ ⟨∅, 𝑥⟩ ∈ V
7		df-inl 9826	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
8	6, 7	dmmpti 6642	. . . 4 ⊢ dom inl = V
9	5, 8	eleqtrrdi 2847	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl)
10		djulcl 9834	. . 3 ⊢ (𝑥 ∈ 𝐴 → (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵))
11	9, 10	jca 511	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴 ⊔ 𝐵)))
12	4, 11	mprgbir 3058	1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ∅c0 4273  {csn 4567  ⟨cop 4573   × cxp 5629  dom cdm 5631   ↾ cres 5633  Fun wfun 6492  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498   ⊔ cdju 9822  inlcinl 9823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1st 7942  df-2nd 7943  df-dju 9825  df-inl 9826

This theorem is referenced by:  inlresf1  9839  updjudhcoinlf  9856  updjud  9858