Theorem f102g 50166

Description: A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)

Assertion

Ref	Expression
f102g	⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶1-1𝐵)

Proof of Theorem f102g

Step	Hyp	Ref	Expression
1		feq2 6707	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵))
2	1	biimpa 482	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵)
3		f0bi 6784	. . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅)
4		f10 6878	. . . . 5 ⊢ ∅:∅⟶1-1𝐵
5		f1eq1 6792	. . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅⟶1-1𝐵 ↔ ∅:∅⟶1-1𝐵))
6	4, 5	mpbiri 261	. . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅⟶1-1𝐵)
7	3, 6	sylbi 220	. . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅⟶1-1𝐵)
8	2, 7	syl 18	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶1-1𝐵)
9		f1eq2 6793	. . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶1-1𝐵 ↔ 𝐹:∅⟶1-1𝐵))
10	9	adantr 486	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴⟶1-1𝐵 ↔ 𝐹:∅⟶1-1𝐵))
11	8, 10	mpbird 260	1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶1-1𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∅c0 4281  ⟶wf 6551  ⟶1-1wf1 6552

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560

This theorem is used by:  f1mo  50167