Theorem alrmomodm 38401

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 3346	. . 3 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
2		brres 5934	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)))
3	2	elv 3441	. . . . 5 ⊢ (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
4		resdm 5974	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	breqd 5100	. . . . 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 2544	. . 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 1921	1 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   ∈ wcel 2111  ∃*wmo 2533  ∃*wrmo 3345  Vcvv 3436   class class class wbr 5089  dom cdm 5614   ↾ cres 5616  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624  df-res 5626

This theorem is referenced by:  inecmo3  38403