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Theorem fnotovb 7483
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6961. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)
Assertion
Ref Expression
fnotovb ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Proof of Theorem fnotovb
StepHypRef Expression
1 fnbrovb 7482 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷𝐹𝑅))
2 df-br 5149 . . . 4 (⟨𝐶, 𝐷𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
32a1i 11 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (⟨𝐶, 𝐷𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
4 df-ot 4640 . . . . . 6 𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅
54eqcomi 2744 . . . . 5 ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑅
65eleq1i 2830 . . . 4 (⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹)
76a1i 11 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
81, 3, 73bitrd 305 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
983impb 1114 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cop 4637  cotp 4639   class class class wbr 5148   × cxp 5687   Fn wfn 6558  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434
This theorem is referenced by: (None)
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