Theorem f1resf1 6813

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

Assertion

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

Proof of Theorem f1resf1

Step	Hyp	Ref	Expression
1		f1ssres 6812	. . 3 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐵)
2	1	3adant3 1150	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐵)
3		frn 6742	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
4	3	3ad2ant3 1153	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
5		f1ssr 6811	. 2 ⊢ (((𝐹 ↾ 𝐶):𝐶⟶1-1𝐵 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐷) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐷)
6	2, 4, 5	syl2anc 596	1 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ⊆ wss 3900  ran crn 5657   ↾ cres 5658  ⟶wf 6557  ⟶1-1wf1 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566

This theorem is used by:  inlresf1  10026  inrresf1  10028  pfxf1  33582  aks6d1c2  43383  3f1oss1  48360