Theorem f1resf1 5552

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 5551	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1ff1 5550	. 2 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)
3	1, 2	sylan 283	1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ⊆ wss 3200   ↾ cres 4727  ⟶wf 5322  –1-1→wf1 5323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331

This theorem is referenced by: (None)