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Theorem funressneu 45692
Description: There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6570. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6878. (Contributed by AV, 7-Sep-2022.)
Assertion
Ref Expression
funressneu (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑉
Allowed substitution hints:   𝐵(𝑦)   𝑊(𝑦)

Proof of Theorem funressneu
StepHypRef Expression
1 simp1l 1198 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴𝑉)
2 simp1r 1199 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐵𝑊)
3 simp3 1139 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴𝐹𝐵)
4 breldmg 5907 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
51, 2, 3, 4syl3anc 1372 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
6 eldmg 5896 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
76ibi 267 . . 3 (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
85, 7syl 17 . 2 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
9 simpl 484 . . . . . 6 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
109anim1i 616 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
11103adant3 1133 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → (𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
12 funressnmo 45691 . . . 4 ((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
1311, 12syl 17 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
14 moeu 2578 . . 3 (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
1513, 14sylib 217 . 2 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
168, 15mpd 15 1 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wex 1782  wcel 2107  ∃*wmo 2533  ∃!weu 2563  {csn 4627   class class class wbr 5147  dom cdm 5675  cres 5677  Fun wfun 6534
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-fun 6542
This theorem is referenced by:  funressnbrafv2  45887
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