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Theorem alrmomorn 36042
 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 3079 . . 3 (∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
2 cnvresrn 36035 . . . . . 6 (𝑅 ↾ ran 𝑅) = 𝑅
32breqi 5036 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥𝑦𝑅𝑥)
4 brres 5828 . . . . . . 7 (𝑥 ∈ V → (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥)))
54elv 3416 . . . . . 6 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑦𝑅𝑥))
6 brcnvg 5717 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 3418 . . . . . . 7 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi2i 626 . . . . . 6 ((𝑦 ∈ ran 𝑅𝑦𝑅𝑥) ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
95, 8bitri 278 . . . . 5 (𝑦(𝑅 ↾ ran 𝑅)𝑥 ↔ (𝑦 ∈ ran 𝑅𝑥𝑅𝑦))
103, 9, 73bitr3i 305 . . . 4 ((𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ 𝑥𝑅𝑦)
1110mobii 2566 . . 3 (∃*𝑦(𝑦 ∈ ran 𝑅𝑥𝑅𝑦) ↔ ∃*𝑦 𝑥𝑅𝑦)
121, 11bitri 278 . 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 209   ∧ wa 400  ∀wal 1537   ∈ wcel 2112  ∃*wmo 2556  ∃*wrmo 3074  Vcvv 3410   class class class wbr 5030  ◡ccnv 5521  ran crn 5523   ↾ cres 5524 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rmo 3079  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-br 5031  df-opab 5093  df-xp 5528  df-rel 5529  df-cnv 5530  df-dm 5532  df-rn 5533  df-res 5534 This theorem is referenced by:  ineccnvmo2  36044
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