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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 6859 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋)) |
3 | 2 | fveq2d 6843 | . 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 512 | . . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
7 | funfvima2 7177 | . . . 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 6859 | . 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
10 | 3, 9 | eqtrd 2776 | 1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
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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