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 6784	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
3		ffvresb 7080	. . 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 3463	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ V)
6		opex 5419	. . . . 5 ⊢ ⟨∅, 𝑥⟩ ∈ V
7		df-inl 9826	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
8	6, 7	dmmpti 6644	. . . 4 ⊢ dom inl = V
9	5, 8	eleqtrrdi 2848	. . 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 3059	1 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  ∅c0 4287  {csn 4582  ⟨cop 4588   × cxp 5630  dom cdm 5632   ↾ cres 5634  Fun wfun 6494  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500   ⊔ 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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7943  df-2nd 7944  df-dju 9825  df-inl 9826

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