Theorem f1ssres 6786

Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Assertion

Ref	Expression
f1ssres	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)

Proof of Theorem f1ssres

Step	Hyp	Ref	Expression
1		f1f 6779	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fssres 6749	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3	1, 2	sylan 580	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4		df-f1 6541	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
5		funres11 6618	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐶))
6	4, 5	simplbiim 504	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶))
7	6	adantr 480	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡(𝐹 ↾ 𝐶))
8		df-f1 6541	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
9	3, 7, 8	sylanbrc 583	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3931  ^◡ccnv 5658   ↾ cres 5661  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541

This theorem is referenced by:  f1resf1  6787  f1ores  6837  oacomf1olem  8581  domssl  9017  undom  9078  pwfseqlem5  10682  hashimarn  14463  hashf1lem2  14479  conjsubgen  19239  sylow1lem2  19585  sylow2blem1  19606  usgrres  29292  lmimdim  33648