Theorem ralmo 38742

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 5890	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑢𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2	1	el3v12 38614	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑥 ∈ ran 𝑅)
3	2	pm4.71ri 566	. . . . 5 ⊢ (𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
4	3	mobii 2554	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
5		moanimv 2625	. . . 4 ⊢ (∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥) ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
6	4, 5	bitri 277	. . 3 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
7	6	albii 1827	. 2 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
8		df-ral 3056	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
9	7, 8	bitr4i 280	1 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   ∈ wcel 2121  ∃*wmo 2543  ∀wral 3055  Vcvv 3433   class class class wbr 5075  ran crn 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by: (None)