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Mirrors > Home > MPE Home > Th. List > ofmresval | Structured version Visualization version GIF version |
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.) |
Ref | Expression |
---|---|
ofmresval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐴) |
ofmresval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
ofmresval | ⊢ (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofmresval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴) | |
2 | ofmresval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
3 | ovres 6698 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐵) → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 × cxp 5036 ↾ cres 5040 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-res 5050 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: psradd 19203 dchrmul 24773 ldualvadd 33434 |
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