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Theorem fnresin 32636
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 6693 . 2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
2 resindi 6013 . . . 4 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵))
3 fnresdm 6687 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
43ineq1d 4219 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹 ∩ (𝐹𝐵)))
5 incom 4209 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹𝐵))
6 resss 6019 . . . . . . 7 (𝐹𝐵) ⊆ 𝐹
7 dfss2 3969 . . . . . . 7 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
86, 7mpbi 230 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
95, 8eqtr3i 2767 . . . . 5 (𝐹 ∩ (𝐹𝐵)) = (𝐹𝐵)
104, 9eqtrdi 2793 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐵))
112, 10eqtrid 2789 . . 3 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) = (𝐹𝐵))
1211fneq1d 6661 . 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 1540  cin 3950  wss 3951  cres 5687   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  fsuppcurry1  32736  fsuppcurry2  32737  signstres  34590
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