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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 6863 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋)) |
3 | 2 | fveq2d 6847 | . 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 513 | . . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
7 | funfvima2 7182 | . . . 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 6863 | . 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
10 | 3, 9 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 dom cdm 5634 ↾ cres 5636 “ cima 5637 Fun wfun 6491 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: wlkres 28660 |
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