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 34466
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 3069 . . 3 (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
2 brresALTV 34375 . . . . . 6 (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥)))
32elv 34328 . . . . 5 (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
4 resdm 5582 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54breqd 4797 . . . . 5 (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥𝑢𝑅𝑥))
63, 5syl5bbr 274 . . . 4 (Rel 𝑅 → ((𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
76mobidv 2639 . . 3 (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
81, 7syl5bb 272 . 2 (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
98albidv 2001 1 (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wcel 2145  ∃*wmo 2619  ∃*wrmo 3064  Vcvv 3351   class class class wbr 4786  dom cdm 5249  cres 5251  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-dm 5259  df-res 5261
This theorem is referenced by:  inecmo3  34468
  Copyright terms: Public domain W3C validator