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Theorem alrmomorn 34365
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 3022 . . 3 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
2 cnvresrn 34358 . . . . . 6 (𝑅 ↾ ran 𝑅) = 𝑅
32breqi 4766 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥𝑦𝑅𝑥)
4 brresALTV 34275 . . . . . . 7 (𝑥 ∈ V → (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥)))
54elv 34228 . . . . . 6 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥))
6 brcnvg 5410 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 34229 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi2i 732 . . . . . 6 ((𝑦 ∈ ran 𝑅𝑦𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
95, 8bitri 264 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
103, 9, 73bitr3i 290 . . . 4 ((𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
1110mobii 2594 . . 3 (∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
121, 11bitri 264 . 2 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
1312albii 1860 1 (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1594  wcel 2103  ∃*wmo 2572  ∃*wrmo 3017  Vcvv 3304   class class class wbr 4760  ccnv 5217  ran crn 5219  cres 5220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rmo 3022  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230
This theorem is referenced by:  ineccnvmo2  34367
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