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Theorem resfvresima 7255
Description: The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)
Hypotheses
Ref Expression
resfvresima.f (𝜑 → Fun 𝐹)
resfvresima.s (𝜑𝑆 ⊆ dom 𝐹)
resfvresima.x (𝜑𝑋𝑆)
Assertion
Ref Expression
resfvresima (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))

Proof of Theorem resfvresima
StepHypRef Expression
1 resfvresima.x . . . 4 (𝜑𝑋𝑆)
21fvresd 6926 . . 3 (𝜑 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
32fveq2d 6910 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)))
4 resfvresima.f . . . . 5 (𝜑 → Fun 𝐹)
5 resfvresima.s . . . . 5 (𝜑𝑆 ⊆ dom 𝐹)
64, 5jca 511 . . . 4 (𝜑 → (Fun 𝐹𝑆 ⊆ dom 𝐹))
7 funfvima2 7251 . . . 4 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
86, 1, 7sylc 65 . . 3 (𝜑 → (𝐹𝑋) ∈ (𝐹𝑆))
98fvresd 6926 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
103, 9eqtrd 2777 1 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  dom cdm 5685  cres 5687  cima 5688  Fun wfun 6555  cfv 6561
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-10 2141  ax-12 2177  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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  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-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  wlkres  29688
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