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Theorem fnotovb 7446
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6935. (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 7445 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷𝐹𝑅))
2 df-br 5145 . . . 4 (⟨𝐶, 𝐷𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
32a1i 11 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (⟨𝐶, 𝐷𝐹𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
4 df-ot 4633 . . . . . 6 𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅
54eqcomi 2742 . . . . 5 ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑅
65eleq1i 2825 . . . 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 1116 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cop 4630  cotp 4632   class class class wbr 5144   × cxp 5670   Fn wfn 6530  (class class class)co 7396
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-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-ot 4633  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fn 6538  df-fv 6543  df-ov 7399
This theorem is referenced by: (None)
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