Theorem ralrnmo 39207

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 5867	. . . . . 6 ⊢ ran 𝑅 = {𝑥 ∣ ∃𝑢 𝑢𝑅𝑥}
2	1	eqabri 2902	. . . . 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 3106	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅(∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
6		df-eu 2594	. . 3 ⊢ (∃!𝑢 𝑢𝑅𝑥 ↔ (∃𝑢 𝑢𝑅𝑥 ∧ ∃*𝑢 𝑢𝑅𝑥))
7	6	ralbii 3108	. 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 2145  ∃*wmo 2562  ∃!weu 2593  ∀wral 3076   class class class wbr 5103  ran crn 5649

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-pr 5391

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 2100  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-cnv 5656  df-dm 5658  df-rn 5659

This theorem is used by: (None)