Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alrmomodm Structured version   Visualization version   GIF version

Theorem alrmomodm 34431
Description: Equivalence of an "at most one" and an "at most one" restricted to the domain inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
alrmomodm (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
Distinct variable groups:   𝑢,𝑅   𝑥,𝑅

Proof of Theorem alrmomodm
StepHypRef Expression
1 df-rmo 3100 . . 3 (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
2 brresALTV 34344 . . . . . 6 (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥)))
32elv 3391 . . . . 5 (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
4 resdm 5640 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54breqd 4848 . . . . 5 (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥𝑢𝑅𝑥))
63, 5syl5bbr 276 . . . 4 (Rel 𝑅 → ((𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
76mobidv 2650 . . 3 (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
81, 7syl5bb 274 . 2 (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
98albidv 2011 1 (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635  wcel 2155  ∃*wmo 2630  ∃*wrmo 3095  Vcvv 3387   class class class wbr 4837  dom cdm 5305  cres 5307  Rel wrel 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rmo 3100  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-br 4838  df-opab 4900  df-xp 5311  df-rel 5312  df-dm 5315  df-res 5317
This theorem is referenced by:  inecmo3  34433
  Copyright terms: Public domain W3C validator