Theorem f102g 49356

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–1-1→𝐵)

Proof of Theorem f102g

Step	Hyp	Ref	Expression
1		feq2 6638	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵))
2	1	biimpa 478	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵)
3		f0bi 6714	. . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅)
4		f10 6804	. . . . 5 ⊢ ∅:∅–1-1→𝐵
5		f1eq1 6722	. . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐵 ↔ ∅:∅–1-1→𝐵))
6	4, 5	mpbiri 260	. . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐵)
7	3, 6	sylbi 219	. . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅–1-1→𝐵)
8	2, 7	syl 17	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅–1-1→𝐵)
9		f1eq2 6723	. . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
10	9	adantr 482	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
11	8, 10	mpbird 259	1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∅c0 4264  ⟶wf 6485  –1-1→wf1 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494

This theorem is referenced by:  f1mo  49357