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Mirrors > Home > ILE Home > Th. List > ovres | GIF version |
Description: The value of a restricted operation. (Contributed by FL, 10-Nov-2006.) |
Ref | Expression |
---|---|
ovres | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4618 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
2 | fvres 5492 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) |
4 | df-ov 5827 | . 2 ⊢ (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) | |
5 | df-ov 5827 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
6 | 3, 4, 5 | 3eqtr4g 2215 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 〈cop 3563 × cxp 4584 ↾ cres 4588 ‘cfv 5170 (class class class)co 5824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-res 4598 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: ovresd 5961 oprssov 5962 ofmresval 6043 elq 9531 xmetres2 12779 blres 12834 xmetresbl 12840 mscl 12865 xmscl 12866 xmsge0 12867 xmseq0 12868 divcnap 12955 cncfmet 12979 |
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