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Theorem mofsn2 46183
Description: There is at most one function into a singleton. An unconditional variant of mofsn 46182, 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 46182 . . . 4 (𝑌 ∈ V → ∃*𝑓 𝑓:𝐴⟶{𝑌})
21adantl 482 . . 3 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶{𝑌})
3 feq3 6592 . . . . 5 (𝐵 = {𝑌} → (𝑓:𝐴𝐵𝑓:𝐴⟶{𝑌}))
43mobidv 2550 . . . 4 (𝐵 = {𝑌} → (∃*𝑓 𝑓:𝐴𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
54adantr 481 . . 3 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → (∃*𝑓 𝑓:𝐴𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
62, 5mpbird 256 . 2 ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴𝐵)
7 simpl 483 . . . 4 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = {𝑌})
8 snprc 4654 . . . . . 6 𝑌 ∈ V ↔ {𝑌} = ∅)
98biimpi 215 . . . . 5 𝑌 ∈ V → {𝑌} = ∅)
109adantl 482 . . . 4 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → {𝑌} = ∅)
117, 10eqtrd 2779 . . 3 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = ∅)
12 mof02 46177 . . 3 (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
1311, 12syl 17 . 2 ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴𝐵)
146, 13pm2.61dan 810 1 (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2107  ∃*wmo 2539  Vcvv 3433  c0 4257  {csn 4562  wf 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-fv 6445
This theorem is referenced by:  mofsssn  46184  mofmo  46185
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