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Theorem fnresin 28812
 Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
fnresin (𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))

Proof of Theorem fnresin
StepHypRef Expression
1 fnresin1 5919 . 2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
2 resindi 5332 . . . 4 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵))
3 fnresdm 5914 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
43ineq1d 3775 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹 ∩ (𝐹𝐵)))
5 incom 3767 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹𝐵))
6 resss 5342 . . . . . . 7 (𝐹𝐵) ⊆ 𝐹
7 df-ss 3554 . . . . . . 7 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
86, 7mpbi 219 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
95, 8eqtr3i 2634 . . . . 5 (𝐹 ∩ (𝐹𝐵)) = (𝐹𝐵)
104, 9syl6eq 2660 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐵))
112, 10syl5eq 2656 . . 3 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) = (𝐹𝐵))
1211fneq1d 5895 . 2 (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵) ↔ (𝐹𝐵) Fn (𝐴𝐵)))
131, 12mpbid 221 1 (𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540   ↾ cres 5040   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806  df-fn 5807 This theorem is referenced by:  signstres  29978
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