Theorem eufsn2 49501

Description: There is exactly one function into a singleton, assuming ax-pow 5334 and ax-un 7730. Variant of eufsn 49500. If existence is not needed, use mofsn 49502 or mofsn2 49503 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)

Hypotheses

Ref	Expression
eufsn.1	⊢ (𝜑 → 𝐵 ∈ 𝑊)
eufsn.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

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

Step	Hyp	Ref	Expression
1		eufsn.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		eufsn.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		snex 5408	. . 3 ⊢ {𝐵} ∈ V
4		xpexg 7745	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V)
5	2, 3, 4	sylancl 597	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V)
6	1, 5	eufsnlem 49499	1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ∈ wcel 2149  ∃!weu 2602  Vcvv 3463  {csn 4591   × cxp 5657  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542

This theorem is referenced by: (None)