Theorem fnresin1 5329

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 3355	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		fnssres 5328	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
3	1, 2	mpan2 425	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∩ cin 3128   ⊆ wss 3129   ↾ cres 4627   Fn wfn 5210

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-res 4637  df-fun 5217  df-fn 5218

This theorem is referenced by:  frecfnom  6399