Theorem fnresin2 5370

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 3381	. 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴
2		fnssres 5368	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))
3	1, 2	mpan2 425	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))

Syntax hints:   → wi 4   ∩ cin 3153   ⊆ wss 3154   ↾ cres 4662   Fn wfn 5250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-res 4672  df-fun 5257  df-fn 5258

This theorem is referenced by: (None)