Theorem mofmo 48808

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 48777	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}))
2		mof02 48800	. . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
3		mofsn2 48806	. . . 4 ⊢ (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
4	3	exlimiv 1930	. . 3 ⊢ (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵)
5	2, 4	jaoi 857	. 2 ⊢ ((𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}) → ∃*𝑓 𝑓:𝐴⟶𝐵)
6	1, 5	sylbi 217	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∅c0 4292  {csn 4585  ⟶wf 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  setcthin  49427