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Theorem funfvop 6927
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
funfvop ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)

Proof of Theorem funfvop
StepHypRef Expression
1 eqid 2738 . 2 (𝐹𝐴) = (𝐹𝐴)
2 funopfvb 6825 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = (𝐹𝐴) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
31, 2mpbii 232 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cop 4567  dom cdm 5589  Fun wfun 6427  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  funfvbrb  6928  fvimacnv  6930  fnopfv  6953  fvelrn  6954  dff3  6976  fnsnb  7038  funfvima3  7112  fprresex  8126  wfrlem17OLD  8156  tfrlem9a  8217  fundmen  8821  adj1  30295  fgreu  31009  fnsnbt  40208
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