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Theorem mofsn2 45686
Description: There is at most one function into a singleton. An unconditional variant of mofsn 45685, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mofsn2 (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑌

Proof of Theorem mofsn2
StepHypRef Expression
1 mofsn 45685 . . . 4 (𝑌 ∈ V → ∃*𝑓 𝑓:𝐴⟶{𝑌})
21adantl 485 . . 3 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶{𝑌})
3 feq3 6481 . . . . 5 (𝐵 = {𝑌} → (𝑓:𝐴𝐵𝑓:𝐴⟶{𝑌}))
43mobidv 2549 . . . 4 (𝐵 = {𝑌} → (∃*𝑓 𝑓:𝐴𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
54adantr 484 . . 3 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → (∃*𝑓 𝑓:𝐴𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
62, 5mpbird 260 . 2 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴𝐵)
7 simpl 486 . . . 4 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = {𝑌})
8 snprc 4605 . . . . . 6 𝑌 ∈ V ↔ {𝑌} = ∅)
98biimpi 219 . . . . 5 𝑌 ∈ V → {𝑌} = ∅)
109adantl 485 . . . 4 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → {𝑌} = ∅)
117, 10eqtrd 2773 . . 3 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = ∅)
12 mof02 45680 . . 3 (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
1311, 12syl 17 . 2 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴𝐵)
146, 13pm2.61dan 813 1 (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  ∃*wmo 2538  Vcvv 3397  c0 4209  {csn 4513  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:  mofsssn  45687  mofmo  45688
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