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Theorem eufsn2 48672
Description: There is exactly one function into a singleton, assuming ax-pow 5370 and ax-un 7753. Variant of eufsn 48671. If existence is not needed, use mofsn 48673 or mofsn2 48674 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 5441 . . 3 {𝐵} ∈ V
4 xpexg 7768 . . 3 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V)
52, 3, 4sylancl 586 . 2 (𝜑 → (𝐴 × {𝐵}) ∈ V)
61, 5eufsnlem 48670 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  ∃!weu 2565  Vcvv 3477  {csn 4630   × cxp 5686  wf 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570
This theorem is referenced by: (None)
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