Theorem mofsn2 49503

Description: There is at most one function into a singleton. An unconditional variant of mofsn 49502, 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:𝐴⟶𝐵)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑌

Proof of Theorem mofsn2

Step	Hyp	Ref	Expression
1		mofsn 49502	. . . 4 ⊢ (𝑌 ∈ V → ∃*𝑓 𝑓:𝐴⟶{𝑌})
2	1	adantl 486	. . 3 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶{𝑌})
3		feq3 6683	. . . . 5 ⊢ (𝐵 = {𝑌} → (𝑓:𝐴⟶𝐵 ↔ 𝑓:𝐴⟶{𝑌}))
4	3	mobidv 2583	. . . 4 ⊢ (𝐵 = {𝑌} → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
5	4	adantr 485	. . 3 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌}))
6	2, 5	mpbird 260	. 2 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶𝐵)
7		simpl 487	. . . 4 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = {𝑌})
8		snprc 4685	. . . . 5 ⊢ (¬ 𝑌 ∈ V ↔ {𝑌} = ∅)
9	8	bilani 509	. . . 4 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → {𝑌} = ∅)
10	7, 9	eqtrd 2804	. . 3 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = ∅)
11		mof02 49497	. . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
12	10, 11	syl 18	. 2 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶𝐵)
13	6, 12	pm2.61dan 824	1 ⊢ (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571  Vcvv 3463  ∅c0 4294  {csn 4591  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542

This theorem is referenced by:  mofsssn  49504  mofmo  49505