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Theorem fnresin 29558
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 6043 . 2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
2 resindi 5447 . . . 4 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵))
3 fnresdm 6038 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
43ineq1d 3846 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹 ∩ (𝐹𝐵)))
5 incom 3838 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹 ∩ (𝐹𝐵))
6 resss 5457 . . . . . . 7 (𝐹𝐵) ⊆ 𝐹
7 df-ss 3621 . . . . . . 7 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
86, 7mpbi 220 . . . . . 6 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
95, 8eqtr3i 2675 . . . . 5 (𝐹 ∩ (𝐹𝐵)) = (𝐹𝐵)
104, 9syl6eq 2701 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐵))
112, 10syl5eq 2697 . . 3 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) = (𝐹𝐵))
1211fneq1d 6019 . 2 (𝐹 Fn 𝐴 → ((𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵) ↔ (𝐹𝐵) Fn (𝐴𝐵)))
131, 12mpbid 222 1 (𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cin 3606  wss 3607  cres 5145   Fn wfn 5921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-fun 5928  df-fn 5929
This theorem is referenced by:  signstres  30780
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