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Theorem fnotovb 5490
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5158. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fnotovb ((𝐹 Fn (A × B) 𝐶 A 𝐷 B) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅 𝐹))

Proof of Theorem fnotovb
StepHypRef Expression
1 opelxpi 4319 . . . 4 ((𝐶 A 𝐷 B) → ⟨𝐶, 𝐷 (A × B))
2 fnopfvb 5158 . . . 4 ((𝐹 Fn (A × B) 𝐶, 𝐷 (A × B)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅 𝐹))
31, 2sylan2 270 . . 3 ((𝐹 Fn (A × B) (𝐶 A 𝐷 B)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅 𝐹))
433impb 1099 . 2 ((𝐹 Fn (A × B) 𝐶 A 𝐷 B) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅 𝐹))
5 df-ov 5458 . . 3 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
65eqeq1i 2044 . 2 ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘⟨𝐶, 𝐷⟩) = 𝑅)
7 df-ot 3377 . . 3 𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅
87eleq1i 2100 . 2 (⟨𝐶, 𝐷, 𝑅 𝐹 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅 𝐹)
94, 6, 83bitr4g 212 1 ((𝐹 Fn (A × B) 𝐶 A 𝐷 B) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cop 3370  cotp 3371   × cxp 4286   Fn wfn 4840  cfv 4845  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-ot 3377  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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