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Theorem resfvresima 6979
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 6669 . . 3 (𝜑 → ((𝐹𝑆)‘𝑋) = (𝐹𝑋))
32fveq2d 6653 . 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 515 . . . 4 (𝜑 → (Fun 𝐹𝑆 ⊆ dom 𝐹))
7 funfvima2 6975 . . . 4 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
86, 1, 7sylc 65 . . 3 (𝜑 → (𝐹𝑋) ∈ (𝐹𝑆))
98fvresd 6669 . 2 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘(𝐹𝑋)) = (𝐻‘(𝐹𝑋)))
103, 9eqtrd 2836 1 (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wss 3884  dom cdm 5523  cres 5525  cima 5526  Fun wfun 6322  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  wlkres  27464
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