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Theorem fnresin 32432
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 6685 . 2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
2 resindi 6005 . . . 4 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵))
3 fnresdm 6679 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
43ineq1d 4213 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹 ∩ (𝐹𝐵)))
5 incom 4203 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹𝐵))
6 resss 6011 . . . . . . 7 (𝐹𝐵) ⊆ 𝐹
7 df-ss 3966 . . . . . . 7 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
86, 7mpbi 229 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
95, 8eqtr3i 2758 . . . . 5 (𝐹 ∩ (𝐹𝐵)) = (𝐹𝐵)
104, 9eqtrdi 2784 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐵))
112, 10eqtrid 2780 . . 3 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) = (𝐹𝐵))
1211fneq1d 6652 . 2 (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵) ↔ (𝐹𝐵) Fn (𝐴𝐵)))
131, 12mpbid 231 1 (𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3948  wss 3949  cres 5684   Fn wfn 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-fun 6555  df-fn 6556
This theorem is referenced by:  fsuppcurry1  32528  fsuppcurry2  32529  signstres  34240
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