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Theorem eufsn2 47895
Description: There is exactly one function into a singleton, assuming ax-pow 5365 and ax-un 7740. Variant of eufsn 47894. If existence is not needed, use mofsn 47896 or mofsn2 47897 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 5433 . . 3 {𝐵} ∈ V
4 xpexg 7752 . . 3 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V)
52, 3, 4sylancl 585 . 2 (𝜑 → (𝐴 × {𝐵}) ∈ V)
61, 5eufsnlem 47893 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  ∃!weu 2558  Vcvv 3471  {csn 4629   × cxp 5676  wf 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556
This theorem is referenced by: (None)
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