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Theorem mofsn 49466
Description: There is at most one function into a singleton, with fewer axioms than eufsn 49464 and eufsn2 49465. See also mofsn2 49467. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mofsn (𝐵𝑉 → ∃*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.
StepHypRef Expression
1 fconst2g 7188 . . . . 5 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
21biimpd 231 . . . 4 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} → 𝑓 = (𝐴 × {𝐵})))
3 fconst2g 7188 . . . . 5 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} ↔ 𝑔 = (𝐴 × {𝐵})))
43biimpd 231 . . . 4 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} → 𝑔 = (𝐴 × {𝐵})))
5 eqtr3 2785 . . . . 5 ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔)
65a1i 11 . . . 4 (𝐵𝑉 → ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔))
72, 4, 6syl2and 617 . . 3 (𝐵𝑉 → ((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
87alrimivv 1949 . 2 (𝐵𝑉 → ∀𝑓𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
9 feq1 6670 . . 3 (𝑓 = 𝑔 → (𝑓:𝐴⟶{𝐵} ↔ 𝑔:𝐴⟶{𝐵}))
109mo4 2594 . 2 (∃*𝑓 𝑓:𝐴⟶{𝐵} ↔ ∀𝑓𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
118, 10sylibr 236 1 (𝐵𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1559   = wceq 1561  wcel 2143  ∃*wmo 2565  {csn 4583   × cxp 5646  wf 6518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fv 6530
This theorem is referenced by:  mofsn2  49467
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