MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnopfv Structured version   Visualization version   GIF version

Theorem fnopfv 7095
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
Assertion
Ref Expression
fnopfv ((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)

Proof of Theorem fnopfv
StepHypRef Expression
1 funfvop 7070 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)
21funfni 6675 1 ((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  cop 4637   Fn wfn 6558  cfv 6563
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-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
This theorem is referenced by:  foeqcnvco  7320
  Copyright terms: Public domain W3C validator