Theorem eufsnlem 49343

Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 49344 assuming ax-rep 5201, or eufsn2 49345 assuming ax-pow 5296 and ax-un 7681. (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 7150	. . . . 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 1935	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
6		eqeq2 2753	. . . . 5 ⊢ (𝑔 = (𝐴 × {𝐵}) → (𝑓 = 𝑔 ↔ 𝑓 = (𝐴 × {𝐵})))
7	6	bibi2d 344	. . . 4 ⊢ (𝑔 = (𝐴 × {𝐵}) → ((𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
8	7	albidv 1928	. . 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 2581	. 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 208  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃!weu 2574  {csn 4557   × cxp 5618  ⟶wf 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496

This theorem is referenced by:  eufsn  49344  eufsn2  49345