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Theorem inlresf 9135
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 9133 . . 3 inl:V–1-1-onto→({∅} × V)
2 f1ofun 6443 . . 3 (inl:V–1-1-onto→({∅} × V) → Fun inl)
3 ffvresb 6709 . . 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 3426 . . . 4 (𝑥𝐴𝑥 ∈ V)
6 opex 5209 . . . . 5 ⟨∅, 𝑥⟩ ∈ V
7 df-inl 9123 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7dmmpti 6319 . . . 4 dom inl = V
95, 8syl6eleqr 2870 . . 3 (𝑥𝐴𝑥 ∈ dom inl)
10 djulcl 9131 . . 3 (𝑥𝐴 → (inl‘𝑥) ∈ (𝐴𝐵))
119, 10jca 504 . 2 (𝑥𝐴 → (𝑥 ∈ dom inl ∧ (inl‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3096 1 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wcel 2051  wral 3081  Vcvv 3408  c0 4172  {csn 4435  cop 4441   × cxp 5401  dom cdm 5403  cres 5405  Fun wfun 6179  wf 6181  1-1-ontowf1o 6184  cfv 6185  cdju 9119  inlcinl 9120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-1st 7499  df-2nd 7500  df-dju 9122  df-inl 9123
This theorem is referenced by:  inlresf1  9136  updjudhcoinlf  9153  updjud  9155
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