Theorem ralmo 38864

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 5919	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑢𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2	1	el3v12 38736	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑥 ∈ ran 𝑅)
3	2	pm4.71ri 568	. . . . 5 ⊢ (𝑢𝑅𝑥 ↔ (𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
4	3	mobii 2577	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑥 ∈ ran 𝑅 ∧ 𝑢𝑅𝑥))
5		moanimv 2648	. . . 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 1841	. 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 280	1 ⊢ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   ∈ wcel 2144  ∃*wmo 2566  ∀wral 3078  Vcvv 3456   class class class wbr 5102  ran crn 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660

This theorem is referenced by: (None)