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Theorem inrresf 9954
Description: The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
inrresf (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)

Proof of Theorem inrresf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 djurf1o 9951 . . 3 inr:V–1-1-onto→({1o} × V)
2 f1ofun 6851 . . 3 (inr:V–1-1-onto→({1o} × V) → Fun inr)
3 ffvresb 7145 . . 3 (Fun inr → ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) ↔ ∀𝑥𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴𝐵))))
41, 2, 3mp2b 10 . 2 ((inr ↾ 𝐵):𝐵⟶(𝐴𝐵) ↔ ∀𝑥𝐵 (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴𝐵)))
5 elex 3499 . . . 4 (𝑥𝐵𝑥 ∈ V)
6 opex 5475 . . . . 5 ⟨1o, 𝑥⟩ ∈ V
7 df-inr 9941 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7dmmpti 6713 . . . 4 dom inr = V
95, 8eleqtrrdi 2850 . . 3 (𝑥𝐵𝑥 ∈ dom inr)
10 djurcl 9949 . . 3 (𝑥𝐵 → (inr‘𝑥) ∈ (𝐴𝐵))
119, 10jca 511 . 2 (𝑥𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3066 1 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wral 3059  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687  dom cdm 5689  cres 5691  Fun wfun 6557  wf 6559  1-1-ontowf1o 6562  cfv 6563  1oc1o 8498  cdju 9936  inrcinr 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-dju 9939  df-inr 9941
This theorem is referenced by:  inrresf1  9955  updjudhcoinrg  9971  updjud  9972
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