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Theorem eufsn2 47757
Description: There is exactly one function into a singleton, assuming ax-pow 5354 and ax-un 7719. Variant of eufsn 47756. If existence is not needed, use mofsn 47758 or mofsn2 47759 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 5422 . . 3 {𝐵} ∈ V
4 xpexg 7731 . . 3 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V)
52, 3, 4sylancl 585 . 2 (𝜑 → (𝐴 × {𝐵}) ∈ V)
61, 5eufsnlem 47755 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  ∃!weu 2554  Vcvv 3466  {csn 4621   × cxp 5665  wf 6530
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542
This theorem is referenced by: (None)
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