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Theorem inrresf 9333
 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 9330 . . 3 inr:V–1-1-onto→({1o} × V)
2 f1ofun 6596 . . 3 (inr:V–1-1-onto→({1o} × V) → Fun inr)
3 ffvresb 6869 . . 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 3462 . . . 4 (𝑥𝐵𝑥 ∈ V)
6 opex 5324 . . . . 5 ⟨1o, 𝑥⟩ ∈ V
7 df-inr 9320 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7dmmpti 6468 . . . 4 dom inr = V
95, 8eleqtrrdi 2904 . . 3 (𝑥𝐵𝑥 ∈ dom inr)
10 djurcl 9328 . . 3 (𝑥𝐵 → (inr‘𝑥) ∈ (𝐴𝐵))
119, 10jca 515 . 2 (𝑥𝐵 → (𝑥 ∈ dom inr ∧ (inr‘𝑥) ∈ (𝐴𝐵)))
124, 11mprgbir 3124 1 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  ∀wral 3109  Vcvv 3444  {csn 4528  ⟨cop 4534   × cxp 5521  dom cdm 5523   ↾ cres 5525  Fun wfun 6322  ⟶wf 6324  –1-1-onto→wf1o 6327  ‘cfv 6328  1oc1o 8082   ⊔ cdju 9315  inrcinr 9317 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1st 7675  df-2nd 7676  df-1o 8089  df-dju 9318  df-inr 9320 This theorem is referenced by:  inrresf1  9334  updjudhcoinrg  9350  updjud  9351
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