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Theorem alrmomorn 36750
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 3352 . . 3 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
2 cnvresrn 36740 . . . . . 6 (𝑅 ↾ ran 𝑅) = 𝑅
32breqi 5110 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥𝑦𝑅𝑥)
4 brres 5941 . . . . . . 7 (𝑥 ∈ V → (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥)))
54elv 3450 . . . . . 6 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥))
6 brcnvg 5832 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 3452 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi2i 624 . . . . . 6 ((𝑦 ∈ ran 𝑅𝑦𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
95, 8bitri 275 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
103, 9, 73bitr3i 301 . . . 4 ((𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
1110mobii 2548 . . 3 (∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
121, 11bitri 275 . 2 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
1312albii 1822 1 (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wal 1540  wcel 2107  ∃*wmo 2538  ∃*wrmo 3351  Vcvv 3444   class class class wbr 5104  ccnv 5630  ran crn 5632  cres 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rmo 3352  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643
This theorem is referenced by:  ineccnvmo2  36752
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