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Theorem alrmomodm 37754
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 3371 . . 3 (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
2 brres 5986 . . . . . 6 (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥)))
32elv 3475 . . . . 5 (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
4 resdm 6024 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54breqd 5153 . . . . 5 (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥𝑢𝑅𝑥))
63, 5bitr3id 285 . . . 4 (Rel 𝑅 → ((𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
76mobidv 2538 . . 3 (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
81, 7bitrid 283 . 2 (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
98albidv 1916 1 (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532  wcel 2099  ∃*wmo 2527  ∃*wrmo 3370  Vcvv 3469   class class class wbr 5142  dom cdm 5672  cres 5674  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rmo 3371  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-dm 5682  df-res 5684
This theorem is referenced by:  inecmo3  37756
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