Theorem alrmomorn 39093

Description: Equivalence of an "at most one" and an "at most one" restricted to the range inside a universal quantification. (Contributed by Peter Mazsa, 3-Sep-2021.)

Assertion

Ref	Expression
alrmomorn	⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)

Proof of Theorem alrmomorn

Step	Hyp	Ref	Expression
1		df-rmo 3367	. . 3 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦))
2		cnvresrn 39083	. . . . . 6 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅
3	2	breqi 5113	. . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ 𝑦◡𝑅𝑥)
4		brres 5983	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥)))
5	4	elv 3458	. . . . . 6 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥))
6		brcnvg 5863	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
7	6	el2v 3460	. . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	anbi2i 635	. . . . . 6 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦))
9	5, 8	bitri 278	. . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦))
10	3, 9, 7	3bitr3i 304	. . . 4 ⊢ ((𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
11	10	mobii 2575	. . 3 ⊢ (∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
12	1, 11	bitri 278	. 2 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
13	12	albii 1852	1 ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∃*wmo 2564  ∃*wrmo 3366  Vcvv 3453   class class class wbr 5107  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661

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-ext 2734  ax-sep 5255  ax-pr 5402

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-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is used by:  ineccnvmo2  39103