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Theorem fnotovb 5460
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5129. (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 4292 . . . 4 ((𝐶 A 𝐷 B) → ⟨𝐶, 𝐷 (A × B))
2 fnopfvb 5129 . . . 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 1081 . 2 ((𝐹 Fn (A × B) 𝐶 A 𝐷 B) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅 𝐹))
5 df-ov 5428 . . 3 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
65eqeq1i 2021 . 2 ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘⟨𝐶, 𝐷⟩) = 𝑅)
7 df-ot 3350 . . 3 𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅
87eleq1i 2077 . 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 867   = wceq 1224   wcel 1367  cop 3343  cotp 3344   × cxp 4259   Fn wfn 4813  cfv 4818  (class class class)co 5425
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-sbc 2734  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-ot 3350  df-uni 3545  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-iota 4783  df-fun 4820  df-fn 4821  df-fv 4826  df-ov 5428
This theorem is referenced by: (None)
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