Theorem eufsn 48830

Description: There is exactly one function into a singleton, assuming ax-rep 5221. See eufsn2 48831 for different axiom requirements. If existence is not needed, use mofsn 48832 or mofsn2 48833 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 5685	. . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		mptexg 7161	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
5	3, 4	eqeltrid 2832	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝐵}) ∈ V)
6	2, 5	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V)
7	1, 6	eufsnlem 48829	1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ∈ wcel 2109  ∃!weu 2561  Vcvv 3438  {csn 4579   ↦ cmpt 5176   × cxp 5621  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by: (None)