Theorem inlresf 9874

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 9872	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1ofun 6805	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
3		ffvresb 7100	. . 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 3471	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V)
6		opex 5427	. . . . 5 ⊢ ⟨∅, 𝑥⟩ ∈ V
7		df-inl 9862	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
8	6, 7	dmmpti 6665	. . . 4 ⊢ dom inl = V
9	5, 8	eleqtrrdi 2840	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl)
10		djulcl 9870	. . 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 3052	1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  Vcvv 3450  ∅c0 4299  {csn 4592  ⟨cop 4598   × cxp 5639  dom cdm 5641   ↾ cres 5643  Fun wfun 6508  ⟶wf 6510  –1-1-onto→wf1o 6513  ‘cfv 6514   ⊔ cdju 9858  inlcinl 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-1st 7971  df-2nd 7972  df-dju 9861  df-inl 9862

This theorem is referenced by:  inlresf1  9875  updjudhcoinlf  9892  updjud  9894