Theorem fssres2 5985

Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)

Assertion

Ref	Expression
fssres2	⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)

Proof of Theorem fssres2

Step	Hyp	Ref	Expression
1		fssres 5983	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵)
2		resabs1 5347	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → ((𝐹 ↾ 𝐴) ↾ 𝐶) = (𝐹 ↾ 𝐶))
3	2	feq1d 5943	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵))
4	3	adantl 481	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵))
5	1, 4	mpbid 221	1 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ⊆ wss 3540   ↾ cres 5040  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808

This theorem is referenced by:  efcvx  24007  filnetlem4  31546