Theorem fnresin2 6664

Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Assertion

Ref	Expression
fnresin2	⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))

Proof of Theorem fnresin2

Step	Hyp	Ref	Expression
1		inss2 4213	. 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴
2		fnssres 6661	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))
3	1, 2	mpan2 691	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))

Syntax hints:   → wi 4   ∩ cin 3925   ⊆ wss 3926   ↾ cres 5656   Fn wfn 6526

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-fun 6533  df-fn 6534

This theorem is referenced by:  resfnfinfin  9349  resfifsupp  9409  hashresfn  14358