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Mirrors > Home > MPE Home > Th. List > ovconst2 | Structured version Visualization version 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 7235 | . 2 ⊢ (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) | |
2 | opelxpi 5603 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → 〈𝑅, 𝑆〉 ∈ (𝐴 × 𝐵)) | |
3 | oprvalconst2.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | 3 | fvconst2 7038 | . . 3 ⊢ (〈𝑅, 𝑆〉 ∈ (𝐴 × 𝐵) → (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) = 𝐶) |
5 | 2, 4 | syl 17 | . 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (((𝐴 × 𝐵) × {𝐶})‘〈𝑅, 𝑆〉) = 𝐶) |
6 | 1, 5 | eqtrid 2790 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3421 {csn 4556 〈cop 4562 × cxp 5564 ‘cfv 6398 (class class class)co 7232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-fv 6406 df-ov 7235 |
This theorem is referenced by: indthinc 46037 indthincALT 46038 |
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