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Mirrors > Home > ILE Home > Th. List > ovconst2 | GIF version |
Description: The value of a constant operation. (Contributed by NM, 5-Nov-2006.) |
Ref | Expression |
---|---|
oprvalconst2.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
ovconst2 | ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5900 | . 2 ⊢ (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) | |
2 | opelxpi 4676 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → 〈𝑅, 𝑆〉 ∈ (𝐴 × 𝐵)) | |
3 | oprvalconst2.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | 3 | fvconst2 5753 | . . 3 ⊢ (〈𝑅, 𝑆〉 ∈ (𝐴 × 𝐵) → (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) = 𝐶) |
5 | 2, 4 | syl 14 | . 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) = 𝐶) |
6 | 1, 5 | eqtrid 2234 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3607 〈cop 3610 × cxp 4642 ‘cfv 5235 (class class class)co 5897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5900 |
This theorem is referenced by: (None) |
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