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Theorem inlresf 9957
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 9955 . . 3 inl:V–1-1-onto→({∅} × V)
2 f1ofun 6845 . . 3 (inl:V–1-1-onto→({∅} × V) → Fun inl)
3 ffvresb 7139 . . 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 3482 . . . 4 (𝑥𝐴𝑥 ∈ V)
6 opex 5470 . . . . 5 ⟨∅, 𝑥⟩ ∈ V
7 df-inl 9945 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7dmmpti 6705 . . . 4 dom inl = V
95, 8eleqtrrdi 2837 . . 3 (𝑥𝐴𝑥 ∈ dom inl)
10 djulcl 9953 . . 3 (𝑥𝐴 → (inl‘𝑥) ∈ (𝐴𝐵))
119, 10jca 510 . 2 (𝑥𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3058 1 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2099  wral 3051  Vcvv 3462  c0 4325  {csn 4633  cop 4639   × cxp 5680  dom cdm 5682  cres 5684  Fun wfun 6548  wf 6550  1-1-ontowf1o 6553  cfv 6554  cdju 9941  inlcinl 9942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-1st 8003  df-2nd 8004  df-dju 9944  df-inl 9945
This theorem is referenced by:  inlresf1  9958  updjudhcoinlf  9975  updjud  9977
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