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Theorem mofmo 45751
Description: There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mofmo (∃*𝑥 𝑥𝐵 → ∃*𝑓 𝑓:𝐴𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,𝑥
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem mofmo
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mo0sn 45740 . 2 (∃*𝑥 𝑥𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
2 mof02 45743 . . 3 (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
3 mofsn2 45749 . . . 4 (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴𝐵)
43exlimiv 1937 . . 3 (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴𝐵)
52, 4jaoi 856 . 2 ((𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}) → ∃*𝑓 𝑓:𝐴𝐵)
61, 5sylbi 220 1 (∃*𝑥 𝑥𝐵 → ∃*𝑓 𝑓:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wex 1786  wcel 2114  ∃*wmo 2539  c0 4221  {csn 4526  wf 6345
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
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 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357
This theorem is referenced by:  setcthin  45847
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