Theorem alrmomodm 39291

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 3366	. . 3 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
2		brres 5977	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)))
3	2	elv 3456	. . . . 5 ⊢ (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))
4		resdm 6015	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	breqd 5114	. . . . 5 ⊢ (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ 𝑢𝑅𝑥))
6	3, 5	bitr3id 288	. . . 4 ⊢ (Rel 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥))
7	6	mobidv 2575	. . 3 ⊢ (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥))
8	1, 7	bitrid 286	. 2 ⊢ (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥))
9	8	albidv 1953	1 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∃*wmo 2563  ∃*wrmo 3365  Vcvv 3451   class class class wbr 5103  dom cdm 5651   ↾ cres 5653  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-res 5663

This theorem is used by:  inecmo3  39301