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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 7429 | . 2 ⊢ (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = (((𝐴 × 𝐵) × {𝐶})‘⟨𝑅, 𝑆⟩) | |
2 | opelxpi 5719 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → ⟨𝑅, 𝑆⟩ ∈ (𝐴 × 𝐵)) | |
3 | oprvalconst2.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | 3 | fvconst2 7222 | . . 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 2780 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ⟨cop 4638 × cxp 5680 ‘cfv 6553 (class class class)co 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 |
This theorem is referenced by: indthinc 48136 indthincALT 48137 |
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