Theorem alrmomodm 38680

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

Step	Hyp	Ref	Expression
1		df-rmo 3342	. . 3 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
2		brres 5951	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)))
3	2	elv 3434	. . . . 5 ⊢ (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
4		resdm 5991	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	breqd 5096	. . . . 5 ⊢ (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ 𝑢𝑅𝑥))
6	3, 5	bitr3id 285	. . . 4 ⊢ (Rel 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
7	6	mobidv 2549	. . 3 ⊢ (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
8	1, 7	bitrid 283	. 2 ⊢ (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
9	8	albidv 1922	1 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∃*wmo 2537  ∃*wrmo 3341  Vcvv 3429   class class class wbr 5085  dom cdm 5631   ↾ cres 5633  Rel wrel 5636

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rmo 3342  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641  df-res 5643

This theorem is referenced by:  inecmo3  38690