Theorem f1mo 48837

Description: A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)

Assertion

Ref	Expression
f1mo	⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem f1mo

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo0sn 48800	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		f102g 48836	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
3		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
4		f1sn2g 48835	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1→𝐵)
5	3, 4	mpan 690	. . . . . 6 ⊢ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵)
6		feq2 6631	. . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:{𝑦}⟶𝐵))
7		f1eq2 6716	. . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:{𝑦}–1-1→𝐵))
8	6, 7	imbi12d 344	. . . . . 6 ⊢ (𝐴 = {𝑦} → ((𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵) ↔ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵)))
9	5, 8	mpbiri 258	. . . . 5 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵))
10	9	exlimiv 1930	. . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵))
11	10	imp 406	. . 3 ⊢ ((∃𝑦 𝐴 = {𝑦} ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
12	2, 11	jaoian 958	. 2 ⊢ (((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
13	1, 12	sylanb 581	1 ⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  Vcvv 3436  ∅c0 4284  {csn 4577  ⟶wf 6478  –1-1→wf1 6479

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 5235  ax-nul 5245  ax-pr 5371

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by:  thincfth  49437