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Theorem alrmomodm 34459
 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 3068 . . 3 (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
2 brresALTV 34368 . . . . . 6 (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥)))
32elv 34321 . . . . 5 (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅𝑢𝑅𝑥))
4 resdm 5582 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54breqd 4795 . . . . 5 (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥𝑢𝑅𝑥))
63, 5syl5bbr 274 . . . 4 (Rel 𝑅 → ((𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
76mobidv 2638 . . 3 (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
81, 7syl5bb 272 . 2 (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
98albidv 2000 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 1628   ∈ wcel 2144  ∃*wmo 2618  ∃*wrmo 3063  Vcvv 3349   class class class wbr 4784  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 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rmo 3068  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-res 5261 This theorem is referenced by:  inecmo3  34461
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