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Theorem ovconst2 7407
Description: The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
Hypothesis
Ref Expression
oprvalconst2.1 𝐶 ∈ V
Assertion
Ref Expression
ovconst2 ((𝑅𝐴𝑆𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶)

Proof of Theorem ovconst2
StepHypRef Expression
1 df-ov 7235 . 2 (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = (((𝐴 × 𝐵) × {𝐶})‘⟨𝑅, 𝑆⟩)
2 opelxpi 5603 . . 3 ((𝑅𝐴𝑆𝐵) → ⟨𝑅, 𝑆⟩ ∈ (𝐴 × 𝐵))
3 oprvalconst2.1 . . . 4 𝐶 ∈ V
43fvconst2 7038 . . 3 (⟨𝑅, 𝑆⟩ ∈ (𝐴 × 𝐵) → (((𝐴 × 𝐵) × {𝐶})‘⟨𝑅, 𝑆⟩) = 𝐶)
52, 4syl 17 . 2 ((𝑅𝐴𝑆𝐵) → (((𝐴 × 𝐵) × {𝐶})‘⟨𝑅, 𝑆⟩) = 𝐶)
61, 5eqtrid 2790 1 ((𝑅𝐴𝑆𝐵) → (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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