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Mirrors > Home > MPE Home > Th. List > resfvresima | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
resfvresima.f | ⊢ (𝜑 → Fun 𝐹) |
resfvresima.s | ⊢ (𝜑 → 𝑆 ⊆ dom 𝐹) |
resfvresima.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Ref | Expression |
---|---|
resfvresima | ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resfvresima.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
2 | 1 | fvresd 6689 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋)) |
3 | 2 | fveq2d 6673 | . 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋))) |
4 | resfvresima.f | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
5 | resfvresima.s | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝐹) | |
6 | 4, 5 | jca 514 | . . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
7 | funfvima2 6992 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))) | |
8 | 6, 1, 7 | sylc 65 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)) |
9 | 8 | fvresd 6689 | . 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
10 | 3, 9 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 dom cdm 5554 ↾ cres 5556 “ cima 5557 Fun wfun 6348 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 |
This theorem is referenced by: wlkres 27451 |
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