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Theorem inlresf 9327
 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⟶(𝐴𝐵)

Proof of Theorem inlresf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 djulf1o 9325 . . 3 inl:V–1-1-onto→({∅} × V)
2 f1ofun 6592 . . 3 (inl:V–1-1-onto→({∅} × V) → Fun inl)
3 ffvresb 6865 . . 3 (Fun inl → ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) ↔ ∀𝑥𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴𝐵))))
41, 2, 3mp2b 10 . 2 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) ↔ ∀𝑥𝐴 (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴𝐵)))
5 elex 3459 . . . 4 (𝑥𝐴𝑥 ∈ V)
6 opex 5321 . . . . 5 ⟨∅, 𝑥⟩ ∈ V
7 df-inl 9315 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7dmmpti 6464 . . . 4 dom inl = V
95, 8eleqtrrdi 2901 . . 3 (𝑥𝐴𝑥 ∈ dom inl)
10 djulcl 9323 . . 3 (𝑥𝐴 → (inl‘𝑥) ∈ (𝐴𝐵))
119, 10jca 515 . 2 (𝑥𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3121 1 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ∅c0 4243  {csn 4525  ⟨cop 4531   × cxp 5517  dom cdm 5519   ↾ cres 5521  Fun wfun 6318  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324   ⊔ cdju 9311  inlcinl 9312 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1st 7671  df-2nd 7672  df-dju 9314  df-inl 9315 This theorem is referenced by:  inlresf1  9328  updjudhcoinlf  9345  updjud  9347
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