Theorem f102g 49478

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 6672	. . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵))
2	1	biimpa 480	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵)
3		f0bi 6749	. . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅)
4		f10 6842	. . . . 5 ⊢ ∅:∅–1-1→𝐵
5		f1eq1 6757	. . . . 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 6758	. . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵))
10	9	adantr 484	. 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 399   = wceq 1562  ∅c0 4287  ⟶wf 6519  –1-1→wf1 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528

This theorem is referenced by:  f1mo  49479