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Theorem alrmomodm 35924
 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 3114 . . 3 (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
2 brres 5829 . . . . . 6 (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥)))
32elv 3447 . . . . 5 (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
4 resdm 5867 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54breqd 5045 . . . . 5 (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥𝑢𝑅𝑥))
63, 5bitr3id 288 . . . 4 (Rel 𝑅 → ((𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
76mobidv 2608 . . 3 (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
81, 7syl5bb 286 . 2 (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
98albidv 1921 1 (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ∈ wcel 2111  ∃*wmo 2596  ∃*wrmo 3109  Vcvv 3442   class class class wbr 5034  dom cdm 5523   ↾ cres 5525  Rel wrel 5528 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rmo 3114  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-dm 5533  df-res 5535 This theorem is referenced by:  inecmo3  35926
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