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Theorem resfvresima 7186
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 6863 . . 3 (𝜑 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
32fveq2d 6847 . 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 513 . . . 4 (𝜑 → (Fun 𝐹𝑆 ⊆ dom 𝐹))
7 funfvima2 7182 . . . 4 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
86, 1, 7sylc 65 . . 3 (𝜑 → (𝐹𝑋) ∈ (𝐹𝑆))
98fvresd 6863 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
103, 9eqtrd 2773 1 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3911  dom cdm 5634  cres 5636  cima 5637  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  wlkres  28660
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