Theorem fnresin1 6617

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

Assertion

Ref	Expression
fnresin1	⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))

Proof of Theorem fnresin1

Step	Hyp	Ref	Expression
1		inss1 4189	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		fnssres 6615	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
3	1, 2	mpan2 691	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∩ cin 3900   ⊆ wss 3901   ↾ cres 5626   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-fun 6494  df-fn 6495

This theorem is referenced by:  frrlem4  8231  fnresin  32702