Theorem f1resf1 6744

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 6743	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2	1	3adant3 1133	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
3		frn 6675	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
4	3	3ad2ant3 1136	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → ran (𝐹 ↾ 𝐶) ⊆ 𝐷)
5		f1ssr 6742	. 2 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)
6	2, 4, 5	syl2anc 585	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)

Syntax hints:   → wi 4   ∧ w3a 1087   ⊆ wss 3889  ran crn 5632   ↾ cres 5633  ⟶wf 6494  –1-1→wf1 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503

This theorem is referenced by:  inlresf1  9839  inrresf1  9841  pfxf1  33002  aks6d1c2  42569  3f1oss1  47523