Theorem mofsn 45685

Description: There is at most one function into a singleton, with fewer axioms than eufsn 45683 and eufsn2 45684. See also mofsn2 45686. (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 6969	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
2	1	biimpd 232	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓:𝐴⟶{𝐵} → 𝑓 = (𝐴 × {𝐵})))
3		fconst2g 6969	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑔:𝐴⟶{𝐵} ↔ 𝑔 = (𝐴 × {𝐵})))
4	3	biimpd 232	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑔:𝐴⟶{𝐵} → 𝑔 = (𝐴 × {𝐵})))
5		eqtr3 2760	. . . . 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 611	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
8	7	alrimivv 1934	. 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑓∀𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
9		feq1 6479	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶{𝐵} ↔ 𝑔:𝐴⟶{𝐵}))
10	9	mo4 2566	. 2 ⊢ (∃*𝑓 𝑓:𝐴⟶{𝐵} ↔ ∀𝑓∀𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
11	8, 10	sylibr 237	1 ⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1540   = wceq 1542   ∈ wcel 2113  ∃*wmo 2538  {csn 4513   × cxp 5517  ⟶wf 6329

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341

This theorem is referenced by:  mofsn2  45686