Theorem f1ssres 6796

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 6788	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fssres 6758	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3	1, 2	sylan 581	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4		df-f1 6549	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
5		funres11 6626	. . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐶))
6	4, 5	simplbiim 506	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶))
7	6	adantr 482	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡(𝐹 ↾ 𝐶))
8		df-f1 6549	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
9	3, 7, 8	sylanbrc 584	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 397   ⊆ wss 3949  ^◡ccnv 5676   ↾ cres 5679  Fun wfun 6538  ⟶wf 6540  –1-1→wf1 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549

This theorem is referenced by:  f1resf1  6797  f1ores  6848  oacomf1olem  8564  domssl  8994  undom  9059  pwfseqlem5  10658  hashimarn  14400  hashf1lem2  14417  conjsubgen  19125  sylow1lem2  19467  sylow2blem1  19488  usgrres  28596  lmimdim  32720