Theorem mofmo 46174

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 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)

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.

Step	Hyp	Ref	Expression
1		mo0sn 46161	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
2		mof02 46166	. . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
3		mofsn2 46172	. . . 4 ⊢ (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
4	3	exlimiv 1933	. . 3 ⊢ (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
5	2, 4	jaoi 854	. 2 ⊢ ((𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}) → ∃*𝑓 𝑓:𝐴⟶𝐵)
6	1, 5	sylbi 216	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃*wmo 2538  ∅c0 4256  {csn 4561  ⟶wf 6429

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

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 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  setcthin  46336