Theorem ralrnmo 39484

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 5876	. . . . . 6 ⊢ ran 𝑅 = {𝑥 ∣ ∃𝑢 𝑢𝑅𝑥}
2	1	eqabri 2906	. . . . 5 ⊢ (𝑥 ∈ ran 𝑅 ↔ ∃𝑢 𝑢𝑅𝑥)
3	2	biimpi 219	. . . 4 ⊢ (𝑥 ∈ ran 𝑅 → ∃𝑢 𝑢𝑅𝑥)
4	3	biantrurd 542	. . 3 ⊢ (𝑥 ∈ ran 𝑅 → (∃*𝑢 𝑢𝑅𝑥 ↔ (∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥)))
5	4	ralbiia 3110	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅(∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
6		df-eu 2597	. . 3 ⊢ (∃!𝑢 𝑢𝑅𝑥 ↔ (∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
7	6	ralbii 3112	. 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 401  ∃wex 1812   ∈ wcel 2147  ∃*wmo 2565  ∃!weu 2596  ∀wral 3080   class class class wbr 5106  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by: (None)