Theorem eufsnlem 47893

Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 47894 assuming ax-rep 5285, or eufsn2 47895 assuming ax-pow 5365 and ax-un 7740. (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 7215	. . . . 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 1923	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
6		eqeq2 2740	. . . . 5 ⊢ (𝑔 = (𝐴 × {𝐵}) → (𝑓 = 𝑔 ↔ 𝑓 = (𝐴 × {𝐵})))
7	6	bibi2d 342	. . . 4 ⊢ (𝑔 = (𝐴 × {𝐵}) → ((𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
8	7	albidv 1916	. . 3 ⊢ (𝑔 = (𝐴 × {𝐵}) → (∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
9	1, 5, 8	spcedv 3585	. 2 ⊢ (𝜑 → ∃𝑔∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔))
10		eu6im 2565	. 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 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃!weu 2558  {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-pr 5429

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-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:  eufsn  47894  eufsn2  47895