MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inrresf Structured version   Visualization version   GIF version

Theorem inrresf 9673
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 9670 . . 3 inr:V–1-1-onto→({1o} × V)
2 f1ofun 6715 . . 3 (inr:V–1-1-onto→({1o} × V) → Fun inr)
3 ffvresb 6993 . . 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 3449 . . . 4 (𝑥𝐵𝑥 ∈ V)
6 opex 5382 . . . . 5 ⟨1o, 𝑥⟩ ∈ V
7 df-inr 9660 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7dmmpti 6574 . . . 4 dom inr = V
95, 8eleqtrrdi 2852 . . 3 (𝑥𝐵𝑥 ∈ dom inr)
10 djurcl 9668 . . 3 (𝑥𝐵 → (inr‘𝑥) ∈ (𝐴𝐵))
119, 10jca 512 . 2 (𝑥𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3081 1 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2110  wral 3066  Vcvv 3431  {csn 4567  cop 4573   × cxp 5587  dom cdm 5589  cres 5591  Fun wfun 6425  wf 6427  1-1-ontowf1o 6430  cfv 6431  1oc1o 8279  cdju 9655  inrcinr 9657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-om 7705  df-1st 7822  df-2nd 7823  df-1o 8286  df-dju 9658  df-inr 9660
This theorem is referenced by:  inrresf1  9674  updjudhcoinrg  9690  updjud  9691
  Copyright terms: Public domain W3C validator