Theorem f1ssres 6583

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 6575	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fssres 6545	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3	1, 2	sylan 583	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4		df-f1 6345	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
5		funres11 6417	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐶))
6	4, 5	simplbiim 508	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶))
7	6	adantr 484	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡(𝐹 ↾ 𝐶))
8		df-f1 6345	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
9	3, 7, 8	sylanbrc 586	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ⊆ wss 3844  ^◡ccnv 5525   ↾ cres 5528  Fun wfun 6334  ⟶wf 6336  –1-1→wf1 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345

This theorem is referenced by:  f1resf1  6584  f1ores  6635  oacomf1olem  8224  pwfseqlem5  10166  hashimarn  13896  hashf1lem2  13911  conjsubgen  18512  sylow1lem2  18845  sylow2blem1  18866  usgrres  27253