Theorem f1resf1 6791

Description: The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)

Assertion

Ref	Expression
f1resf1	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)

Proof of Theorem f1resf1

Step	Hyp	Ref	Expression
1		f1ssres 6790	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2	1	3adant3 1150	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
3		frn 6720	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
4	3	3ad2ant3 1153	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
5		f1ssr 6789	. 2 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)
6	2, 4, 5	syl2anc 596	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ⊆ wss 3908  ran crn 5667   ↾ cres 5668  ⟶wf 6539  –1-1→wf1 6540

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548

This theorem is used by:  inlresf1  9920  inrresf1  9922  pfxf1  33299  aks6d1c2  42938  3f1oss1  47853