Theorem alrmomorn 38897

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 3376	. . 3 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦))
2		cnvresrn 38887	. . . . . 6 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅
3	2	breqi 5119	. . . . 5 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ 𝑦◡𝑅𝑥)
4		brres 5986	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥)))
5	4	elv 3468	. . . . . 6 ⊢ (𝑦(◡𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅 ∧ 𝑦◡𝑅𝑥))
6		brcnvg 5866	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
7	6	el2v 3470	. . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	anbi2i 634	. . . . . 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 2582	. . 3 ⊢ (∃*𝑦(𝑦 ∈ ran 𝑅 ∧ 𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
12	1, 11	bitri 278	. 2 ⊢ (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
13	12	albii 1846	1 ⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1565   ∈ wcel 2149  ∃*wmo 2571  ∃*wrmo 3375  Vcvv 3463   class class class wbr 5113  ^◡ccnv 5661  ran crn 5663   ↾ cres 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rmo 3376  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674

This theorem is referenced by:  ineccnvmo2  38907