Theorem alrmomodm 38317

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 3388	. . 3 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
2		brres 6018	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)))
3	2	elv 3493	. . . . 5 ⊢ (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
4		resdm 6057	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	breqd 5177	. . . . 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 2552	. . 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 1919	1 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   ∈ wcel 2108  ∃*wmo 2541  ∃*wrmo 3387  Vcvv 3488   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-res 5712

This theorem is referenced by:  inecmo3  38319