Theorem f1ssres 6817

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⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐵)

Proof of Theorem f1ssres

Step	Hyp	Ref	Expression
1		f1f 6808	. . 3 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶𝐵)
2		fssres 6778	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3	1, 2	sylan 592	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4		df-f1 6571	. . . 4 ⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
5		funres11 6647	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐶))
6	4, 5	simplbiim 514	. . 3 ⊢ (𝐹:𝐴⟶1-1𝐵 → Fun ◡(𝐹 ↾ 𝐶))
7	6	adantr 486	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡(𝐹 ↾ 𝐶))
8		df-f1 6571	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶1-1𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
9	3, 7, 8	sylanbrc 595	1 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶1-1𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3900  ^◡ccnv 5657   ↾ cres 5660  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563

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 5255  ax-pr 5398

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 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571

This theorem is used by:  f1resf1  6818  f1ores  6869  oacomf1olem  8610  domssl  9063  undom  9122  pwfseqlem5  10787  hashimarn  14626  hashf1lem2  14642  conjsubgen  19515  sylow1lem2  19864  sylow2blem1  19885  usgrres  29946  lmimdim  34347  vonf1oonf1  36084  onprcf1of1orad  36094  onprcf1of1orbd  36095