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Theorem alrmomorn 35493
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
StepHypRef Expression
1 df-rmo 3143 . . 3 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
2 cnvresrn 35486 . . . . . 6 (𝑅 ↾ ran 𝑅) = 𝑅
32breqi 5063 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥𝑦𝑅𝑥)
4 brres 5853 . . . . . . 7 (𝑥 ∈ V → (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥)))
54elv 3497 . . . . . 6 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥))
6 brcnvg 5743 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 3499 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi2i 622 . . . . . 6 ((𝑦 ∈ ran 𝑅𝑦𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
95, 8bitri 276 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
103, 9, 73bitr3i 302 . . . 4 ((𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
1110mobii 2624 . . 3 (∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
121, 11bitri 276 . 2 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
1312albii 1811 1 (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wal 1526  wcel 2105  ∃*wmo 2613  ∃*wrmo 3138  Vcvv 3492   class class class wbr 5057  ccnv 5547  ran crn 5549  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rmo 3143  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by:  ineccnvmo2  35495
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