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Theorem mofsn 49403
Description: There is at most one function into a singleton, with fewer axioms than eufsn 49401 and eufsn2 49402. See also mofsn2 49404. (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 7172 . . . . 5 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
21biimpd 231 . . . 4 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} → 𝑓 = (𝐴 × {𝐵})))
3 fconst2g 7172 . . . . 5 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} ↔ 𝑔 = (𝐴 × {𝐵})))
43biimpd 231 . . . 4 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} → 𝑔 = (𝐴 × {𝐵})))
5 eqtr3 2774 . . . . 5 ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔)
65a1i 11 . . . 4 (𝐵𝑉 → ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔))
72, 4, 6syl2and 616 . . 3 (𝐵𝑉 → ((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
87alrimivv 1938 . 2 (𝐵𝑉 → ∀𝑓𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
9 feq1 6654 . . 3 (𝑓 = 𝑔 → (𝑓:𝐴⟶{𝐵} ↔ 𝑔:𝐴⟶{𝐵}))
109mo4 2583 . 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 398  wal 1548   = wceq 1550  wcel 2132  ∃*wmo 2554  {csn 4572   × cxp 5634  wf 6502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fv 6514
This theorem is referenced by:  mofsn2  49404
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