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Theorem fnresin 32643
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 6694 . 2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
2 resindi 6016 . . . 4 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵))
3 fnresdm 6688 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
43ineq1d 4227 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹 ∩ (𝐹𝐵)))
5 incom 4217 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹𝐵))
6 resss 6022 . . . . . . 7 (𝐹𝐵) ⊆ 𝐹
7 dfss2 3981 . . . . . . 7 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
86, 7mpbi 230 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
95, 8eqtr3i 2765 . . . . 5 (𝐹 ∩ (𝐹𝐵)) = (𝐹𝐵)
104, 9eqtrdi 2791 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐵))
112, 10eqtrid 2787 . . 3 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) = (𝐹𝐵))
1211fneq1d 6662 . 2 (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵) ↔ (𝐹𝐵) Fn (𝐴𝐵)))
131, 12mpbid 232 1 (𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  wss 3963  cres 5691   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566
This theorem is referenced by:  fsuppcurry1  32743  fsuppcurry2  32744  signstres  34569
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