Theorem eufsnlem 49316

Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 49317 assuming ax-rep 5212, or eufsn2 49318 assuming ax-pow 5307 and ax-un 7689. (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 7158	. . . . 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 1929	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
6		eqeq2 2748	. . . . 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 1922	. . 3 ⊢ (𝑔 = (𝐴 × {𝐵}) → (∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
9	1, 5, 8	spcedv 3540	. 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 206  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2568  {csn 4567   × cxp 5629  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  eufsn  49317  eufsn2  49318