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Theorem alrmomorn 34438
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 3111 . . 3 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
2 cnvresrn 34431 . . . . . 6 (𝑅 ↾ ran 𝑅) = 𝑅
32breqi 4857 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥𝑦𝑅𝑥)
4 brresALTV 34352 . . . . . . 7 (𝑥 ∈ V → (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥)))
54elv 3402 . . . . . 6 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥))
6 brcnvg 5511 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 34308 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi2i 611 . . . . . 6 ((𝑦 ∈ ran 𝑅𝑦𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
95, 8bitri 266 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
103, 9, 73bitr3i 292 . . . 4 ((𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
1110mobii 2662 . . 3 (∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
121, 11bitri 266 . 2 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦 𝑥𝑅𝑦)
1312albii 1904 1 (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wal 1635  wcel 2157  ∃*wmo 2633  ∃*wrmo 3106  Vcvv 3398   class class class wbr 4851  ccnv 5317  ran crn 5319  cres 5320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rmo 3111  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324  df-rel 5325  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330
This theorem is referenced by:  ineccnvmo2  34440
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