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Theorem funressneu 47032
Description: There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6589. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6894. (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 5918 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
51, 2, 3, 4syl3anc 1373 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
6 eldmg 5907 . . . 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 482 . . . . . 6 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
109anim1i 615 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
11103adant3 1133 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → (𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
12 funressnmo 47031 . . . 4 ((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
1311, 12syl 17 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
14 moeu 2582 . . 3 (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
1513, 14sylib 218 . 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 395  w3a 1087  wex 1779  wcel 2108  ∃*wmo 2537  ∃!weu 2567  {csn 4624   class class class wbr 5141  dom cdm 5683  cres 5685  Fun wfun 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-res 5695  df-fun 6561
This theorem is referenced by:  funressnbrafv2  47229
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