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Mirrors > Home > ILE Home > Th. List > ofmresval | 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 5660 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐵) → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) | |
4 | 1, 2, 3 | syl2anc 403 | 1 ⊢ (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 × cxp 4361 ↾ cres 4365 (class class class)co 5532 ∘𝑓 cof 5730 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-res 4375 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: (None) |
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