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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 6926 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋)) |
| 3 | 2 | fveq2d 6910 | . 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 511 | . . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
| 7 | funfvima2 7251 | . . . 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 6926 | . 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
| 10 | 3, 9 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: wlkres 29688 |
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