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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 7364 | . 2 ⊢ (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = (((𝐴 × 𝐵) × {𝐶})‘⟨𝑅, 𝑆⟩) | |
2 | opelxpi 5674 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → ⟨𝑅, 𝑆⟩ ∈ (𝐴 × 𝐵)) | |
3 | oprvalconst2.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | 3 | fvconst2 7157 | . . 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 2785 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 {csn 4590 ⟨cop 4596 × cxp 5635 ‘cfv 6500 (class class class)co 7361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 |
This theorem is referenced by: indthinc 47162 indthincALT 47163 |
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