Theorem ralmo 38701

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 5892	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑢𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2	1	el3v12 38573	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑥 ∈ ran 𝑅)
3	2	pm4.71ri 560	. . . . 5 ⊢ (𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
4	3	mobii 2549	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
5		moanimv 2620	. . . 4 ⊢ (∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥) ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
6	4, 5	bitri 275	. . 3 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
7	6	albii 1821	. 2 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
8		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ ran 𝑅 → ∃*𝑢 𝑢𝑅𝑥))
9	7, 8	bitr4i 278	1 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  Vcvv 3430   class class class wbr 5086  ran crn 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5634  df-dm 5636  df-rn 5637

This theorem is referenced by: (None)