Theorem f1mo 49781

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 49744	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		f102g 49780	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
3		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
4		f1sn2g 49779	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1→𝐵)
5	3, 4	mpan 703	. . . . . 6 ⊢ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵)
6		feq2 6681	. . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:{𝑦}⟶𝐵))
7		f1eq2 6767	. . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:{𝑦}–1-1→𝐵))
8	6, 7	imbi12d 347	. . . . . 6 ⊢ (𝐴 = {𝑦} → ((𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵) ↔ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵)))
9	5, 8	mpbiri 261	. . . . 5 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵))
10	9	exlimiv 1963	. . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵))
11	10	imp 412	. . 3 ⊢ ((∃𝑦 𝐴 = {𝑦} ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
12	2, 11	jaoian 971	. 2 ⊢ (((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
13	1, 12	sylanb 593	1 ⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2562  Vcvv 3450  ∅c0 4279  {csn 4584  ⟶wf 6529  –1-1→wf1 6530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is used by:  thincfth  50378