Theorem eufsn 49332

Description: There is exactly one function into a singleton, assuming ax-rep 5199. See eufsn2 49333 for different axiom requirements. If existence is not needed, use mofsn 49334 or mofsn2 49335 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
eufsn	⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

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

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

Proof of Theorem eufsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufsn.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		eufsn.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fconstmpt 5680	. . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		mptexg 7165	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
5	3, 4	eqeltrid 2843	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝐵}) ∈ V)
6	2, 5	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V)
7	1, 6	eufsnlem 49331	1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ∈ wcel 2119  ∃!weu 2572  Vcvv 3431  {csn 4555   ↦ cmpt 5153   × cxp 5616  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493

This theorem is referenced by: (None)