Theorem inlresf 9826

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 9824	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1ofun 6776	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
3		ffvresb 7070	. . 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 3461	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V)
6		opex 5412	. . . . 5 ⊢ ⟨∅, 𝑥⟩ ∈ V
7		df-inl 9814	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
8	6, 7	dmmpti 6636	. . . 4 ⊢ dom inl = V
9	5, 8	eleqtrrdi 2847	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ dom inl)
10		djulcl 9822	. . 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 2113  ∀wral 3051  Vcvv 3440  ∅c0 4285  {csn 4580  ⟨cop 4586   × cxp 5622  dom cdm 5624   ↾ cres 5626  Fun wfun 6486  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492   ⊔ cdju 9810  inlcinl 9811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7933  df-2nd 7934  df-dju 9813  df-inl 9814

This theorem is referenced by:  inlresf1  9827  updjudhcoinlf  9844  updjud  9846