Theorem f102g 49550

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 6685	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵))
2	1	biimpa 481	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵)
3		f0bi 6762	. . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅)
4		f10 6855	. . . . 5 ⊢ ∅:∅–1-1→𝐵
5		f1eq1 6770	. . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐵 ↔ ∅:∅–1-1→𝐵))
6	4, 5	mpbiri 261	. . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐵)
7	3, 6	sylbi 220	. . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅–1-1→𝐵)
8	2, 7	syl 18	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅–1-1→𝐵)
9		f1eq2 6771	. . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
10	9	adantr 485	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
11	8, 10	mpbird 260	1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∅c0 4292  ⟶wf 6533  –1-1→wf1 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542

This theorem is referenced by:  f1mo  49551