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Theorem resfvresima 7181
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 6859 . . 3 (𝜑 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
32fveq2d 6843 . 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 512 . . . 4 (𝜑 → (Fun 𝐹𝑆 ⊆ dom 𝐹))
7 funfvima2 7177 . . . 4 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
86, 1, 7sylc 65 . . 3 (𝜑 → (𝐹𝑋) ∈ (𝐹𝑆))
98fvresd 6859 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
103, 9eqtrd 2776 1 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wss 3908  dom cdm 5631  cres 5633  cima 5634  Fun wfun 6487  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501
This theorem is referenced by:  wlkres  28504
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