Theorem ralmo 38727

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 5883	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑢𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2	1	el3v12 38599	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑥 ∈ ran 𝑅)
3	2	pm4.71ri 565	. . . . 5 ⊢ (𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
4	3	mobii 2552	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
5		moanimv 2623	. . . 4 ⊢ (∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥) ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
6	4, 5	bitri 276	. . 3 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
7	6	albii 1826	. 2 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
8		df-ral 3054	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
9	7, 8	bitr4i 279	1 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3053  Vcvv 3431   class class class wbr 5072  ran crn 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629

This theorem is referenced by: (None)