Theorem ralmo 39095

Description: "At most one" can be restricted to the range. (Contributed by Peter Mazsa, 2-Feb-2026.)

Assertion

Ref	Expression
ralmo	⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

Distinct variable groups:   𝑢,𝑅   𝑥,𝑢

Allowed substitution hint:   𝑅(𝑥)

Proof of Theorem ralmo

Step	Hyp	Ref	Expression
1		brelrng 5929	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑢𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2	1	el3v12 38967	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑥 ∈ ran 𝑅)
3	2	pm4.71ri 570	. . . . 5 ⊢ (𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
4	3	mobii 2575	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
5		moanimv 2646	. . . 4 ⊢ (∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥) ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
6	4, 5	bitri 278	. . 3 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
7	6	albii 1852	. 2 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
8		df-ral 3079	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
9	7, 8	bitr4i 281	1 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∃*wmo 2564  ∀wral 3078  Vcvv 3453   class class class wbr 5107  ran crn 5660

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 2734  ax-sep 5255  ax-pr 5402

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 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is used by: (None)