Theorem ralrnmo 39038

Description: On the range, "at most one" becomes "exactly one". (Contributed by Peter Mazsa, 27-Sep-2018.) (Revised by Peter Mazsa, 2-Feb-2026.)

Assertion

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

Distinct variable group:   𝑢,𝑅,𝑥

Proof of Theorem ralrnmo

Step	Hyp	Ref	Expression
1		dfrn2 5877	. . . . . 6 ⊢ ran 𝑅 = {𝑥 ∣ ∃𝑢 𝑢𝑅𝑥}
2	1	eqabri 2904	. . . . 5 ⊢ (𝑥 ∈ ran 𝑅 ↔ ∃𝑢 𝑢𝑅𝑥)
3	2	biimpi 219	. . . 4 ⊢ (𝑥 ∈ ran 𝑅 → ∃𝑢 𝑢𝑅𝑥)
4	3	biantrurd 541	. . 3 ⊢ (𝑥 ∈ ran 𝑅 → (∃*𝑢 𝑢𝑅𝑥 ↔ (∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥)))
5	4	ralbiia 3108	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅(∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
6		df-eu 2596	. . 3 ⊢ (∃!𝑢 𝑢𝑅𝑥 ↔ (∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
7	6	ralbii 3110	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃!𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅(∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
8	5, 7	bitr4i 281	1 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃!𝑢 𝑢𝑅𝑥)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400  ∃wex 1808   ∈ wcel 2142  ∃*wmo 2564  ∃!weu 2595  ∀wral 3078   class class class wbr 5108  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by: (None)