Theorem mofsn 48832

Description: There is at most one function into a singleton, with fewer axioms than eufsn 48830 and eufsn2 48831. See also mofsn2 48833. (Contributed by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
mofsn	⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})

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

Proof of Theorem mofsn

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconst2g 7177	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
2	1	biimpd 229	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓:𝐴⟶{𝐵} → 𝑓 = (𝐴 × {𝐵})))
3		fconst2g 7177	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑔:𝐴⟶{𝐵} ↔ 𝑔 = (𝐴 × {𝐵})))
4	3	biimpd 229	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑔:𝐴⟶{𝐵} → 𝑔 = (𝐴 × {𝐵})))
5		eqtr3 2751	. . . . 5 ⊢ ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔)
6	5	a1i 11	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔))
7	2, 4, 6	syl2and 608	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
8	7	alrimivv 1928	. 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑓∀𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
9		feq1 6666	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶{𝐵} ↔ 𝑔:𝐴⟶{𝐵}))
10	9	mo4 2559	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶{𝐵} ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
11	8, 10	sylibr 234	1 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  {csn 4589   × cxp 5636  ⟶wf 6507

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-sep 5251  ax-nul 5261  ax-pr 5387

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-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  mofsn2  48833