Theorem mofmo 50066

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 50035	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
2		mof02 50058	. . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
3		mofsn2 50064	. . . 4 ⊢ (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
4	3	exlimiv 1962	. . 3 ⊢ (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
5	2, 4	jaoi 871	. 2 ⊢ ((𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}) → ∃*𝑓 𝑓:𝐴⟶𝐵)
6	1, 5	sylbi 220	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  ∅c0 4277  {csn 4584  ⟶wf 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is used by:  setcthin  50708