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Theorem eufsn2 48752
Description: There is exactly one function into a singleton, assuming ax-pow 5365 and ax-un 7755. Variant of eufsn 48751. If existence is not needed, use mofsn 48753 or mofsn2 48754 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
Hypotheses
Ref Expression
eufsn.1 (𝜑𝐵𝑊)
eufsn.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
eufsn2 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem eufsn2
StepHypRef Expression
1 eufsn.1 . 2 (𝜑𝐵𝑊)
2 eufsn.2 . . 3 (𝜑𝐴𝑉)
3 snex 5436 . . 3 {𝐵} ∈ V
4 xpexg 7770 . . 3 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V)
52, 3, 4sylancl 586 . 2 (𝜑 → (𝐴 × {𝐵}) ∈ V)
61, 5eufsnlem 48750 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  ∃!weu 2568  Vcvv 3480  {csn 4626   × cxp 5683  wf 6557
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by: (None)
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