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Theorem funressnmo 46333
Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
funressnmo ((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑉

Proof of Theorem funressnmo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4633 . . . . . 6 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21reseq2d 5975 . . . . 5 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
32funeqd 6564 . . . 4 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
4 breq1 5144 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
54mobidv 2537 . . . 4 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
63, 5imbi12d 344 . . 3 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦)))
7 funressnvmo 46332 . . 3 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
86, 7vtoclg 3537 . 2 (𝐴𝑉 → (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦))
98imp 406 1 ((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ∃*wmo 2526  {csn 4623   class class class wbr 5141  cres 5671  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-res 5681  df-fun 6539
This theorem is referenced by:  funressneu  46334
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