Theorem inlresf 9983

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 9981	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1ofun 6864	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
3		ffvresb 7159	. . 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 3509	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V)
6		opex 5484	. . . . 5 ⊢ ⟨∅, 𝑥⟩ ∈ V
7		df-inl 9971	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
8	6, 7	dmmpti 6724	. . . 4 ⊢ dom inl = V
9	5, 8	eleqtrrdi 2855	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl)
10		djulcl 9979	. . 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 3074	1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ∅c0 4352  {csn 4648  ⟨cop 4654   × cxp 5698  dom cdm 5700   ↾ cres 5702  Fun wfun 6567  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573   ⊔ cdju 9967  inlcinl 9968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031  df-dju 9970  df-inl 9971

This theorem is referenced by:  inlresf1  9984  updjudhcoinlf  10001  updjud  10003