Theorem eufsnlem 46168

Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 46169 assuming ax-rep 5209, or eufsn2 46170 assuming ax-pow 5288 and ax-un 7588. (Contributed by Zhi Wang, 19-Sep-2024.)

Hypotheses

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

Assertion

Ref	Expression
eufsnlem	⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓

Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem eufsnlem

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufsnlem.2	. . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)
2		eufsn.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		fconst2g 7078	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
5	4	alrimiv 1930	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
6		eqeq2 2750	. . . . 5 ⊢ (𝑔 = (𝐴 × {𝐵}) → (𝑓 = 𝑔 ↔ 𝑓 = (𝐴 × {𝐵})))
7	6	bibi2d 343	. . . 4 ⊢ (𝑔 = (𝐴 × {𝐵}) → ((𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
8	7	albidv 1923	. . 3 ⊢ (𝑔 = (𝐴 × {𝐵}) → (∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
9	1, 5, 8	spcedv 3537	. 2 ⊢ (𝜑 → ∃𝑔∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔))
10		eu6im 2575	. 2 ⊢ (∃𝑔∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) → ∃!𝑓 𝑓:𝐴⟶{𝐵})
11	9, 10	syl 17	1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃!weu 2568  {csn 4561   × cxp 5587  ⟶wf 6429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  eufsn  46169  eufsn2  46170