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Theorem fnopafv2b 47261
Description: Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
fnopafv2b ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Proof of Theorem fnopafv2b
StepHypRef Expression
1 fnbrafv2b 47260 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
2 df-br 5144 . 2 (𝐵𝐹𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹)
31, 2bitrdi 287 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143   Fn wfn 6556  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  funopafv2b  47263
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