Theorem f1mo 48566

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 48547	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		f102g 48565	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
3		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
4		f1sn2g 48564	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1→𝐵)
5	3, 4	mpan 689	. . . . . 6 ⊢ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵)
6		feq2 6729	. . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:{𝑦}⟶𝐵))
7		f1eq2 6813	. . . . . . 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 1929	. . . 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 957	. 2 ⊢ (((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
13	1, 12	sylanb 580	1 ⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  Vcvv 3488  ∅c0 4352  {csn 4648  ⟶wf 6569  –1-1→wf1 6570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  thincfth  48715