Theorem fnresin2 6661

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 4190	. 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴
2		fnssres 6658	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))
3	1, 2	mpan2 703	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3904   ⊆ wss 3905   ↾ cres 5663   Fn wfn 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-fun 6538  df-fn 6539

This theorem is used by:  resfnfinfin  9290  resfifsupp  9353  hashresfn  14381