Theorem f102g 46179

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 6582	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵))
2	1	biimpa 477	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵)
3		f0bi 6657	. . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅)
4		f10 6749	. . . . 5 ⊢ ∅:∅–1-1→𝐵
5		f1eq1 6665	. . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐵 ↔ ∅:∅–1-1→𝐵))
6	4, 5	mpbiri 257	. . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐵)
7	3, 6	sylbi 216	. . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅–1-1→𝐵)
8	2, 7	syl 17	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅–1-1→𝐵)
9		f1eq2 6666	. . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
10	9	adantr 481	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
11	8, 10	mpbird 256	1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∅c0 4256  ⟶wf 6429  –1-1→wf1 6430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438

This theorem is referenced by:  f1mo  46180