Theorem ralrnmo 38865

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 5866	. . . . . 6 ⊢ ran 𝑅 = {𝑥 ∣ ∃𝑢 𝑢𝑅𝑥}
2	1	eqabri 2906	. . . . 5 ⊢ (𝑥 ∈ ran 𝑅 ↔ ∃𝑢 𝑢𝑅𝑥)
3	2	biimpi 218	. . . 4 ⊢ (𝑥 ∈ ran 𝑅 → ∃𝑢 𝑢𝑅𝑥)
4	3	biantrurd 540	. . 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 2598	. . 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 280	1 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃!𝑢 𝑢𝑅𝑥)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1801   ∈ wcel 2144  ∃*wmo 2566  ∃!weu 2597  ∀wral 3078   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-12 2214  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-eu 2598  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)