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Theorem resfvresima 6534
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)
2 fvres 6245 . . . 4 (𝑋𝑆 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
31, 2syl 17 . . 3 (𝜑 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
43fveq2d 6233 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)))
5 resfvresima.f . . . . 5 (𝜑 → Fun 𝐹)
6 resfvresima.s . . . . 5 (𝜑𝑆 ⊆ dom 𝐹)
75, 6jca 553 . . . 4 (𝜑 → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 funfvima2 6533 . . . 4 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
97, 1, 8sylc 65 . . 3 (𝜑 → (𝐹𝑋) ∈ (𝐹𝑆))
10 fvres 6245 . . 3 ((𝐹𝑋) ∈ (𝐹𝑆) → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
119, 10syl 17 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
124, 11eqtrd 2685 1 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wss 3607  dom cdm 5143  cres 5145  cima 5146  Fun wfun 5920  cfv 5926
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-eu 2502  df-mo 2503  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-sbc 3469  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-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  wlkres  26623
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