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Theorem mofsn 47599
Description: There is at most one function into a singleton, with fewer axioms than eufsn 47597 and eufsn2 47598. See also mofsn2 47600. (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 7207 . . . . 5 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
21biimpd 228 . . . 4 (𝐵𝑉 → (𝑓:𝐴⟶{𝐵} → 𝑓 = (𝐴 × {𝐵})))
3 fconst2g 7207 . . . . 5 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} ↔ 𝑔 = (𝐴 × {𝐵})))
43biimpd 228 . . . 4 (𝐵𝑉 → (𝑔:𝐴⟶{𝐵} → 𝑔 = (𝐴 × {𝐵})))
5 eqtr3 2756 . . . . 5 ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔)
65a1i 11 . . . 4 (𝐵𝑉 → ((𝑓 = (𝐴 × {𝐵}) ∧ 𝑔 = (𝐴 × {𝐵})) → 𝑓 = 𝑔))
72, 4, 6syl2and 606 . . 3 (𝐵𝑉 → ((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
87alrimivv 1929 . 2 (𝐵𝑉 → ∀𝑓𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
9 feq1 6699 . . 3 (𝑓 = 𝑔 → (𝑓:𝐴⟶{𝐵} ↔ 𝑔:𝐴⟶{𝐵}))
109mo4 2558 . 2 (∃*𝑓 𝑓:𝐴⟶{𝐵} ↔ ∀𝑓𝑔((𝑓:𝐴⟶{𝐵} ∧ 𝑔:𝐴⟶{𝐵}) → 𝑓 = 𝑔))
118, 10sylibr 233 1 (𝐵𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1537   = wceq 1539  wcel 2104  ∃*wmo 2530  {csn 4629   × cxp 5675  wf 6540
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  mofsn2  47600
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