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Theorem funressnmo 42687
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 4445 . . . . . 6 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21reseq2d 5689 . . . . 5 (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
32funeqd 6204 . . . 4 (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
4 breq1 4926 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
54mobidv 2561 . . . 4 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
63, 5imbi12d 337 . . 3 (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦)))
7 funressnvmo 42685 . . 3 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
86, 7vtoclg 3480 . 2 (𝐴𝑉 → (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦))
98imp 398 1 ((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  ∃*wmo 2545  {csn 4435   class class class wbr 4923  cres 5403  Fun wfun 6176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-res 5413  df-fun 6184
This theorem is referenced by:  funressneu  42688
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