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Theorem funfvop 6295
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 2621 . 2 (𝐹𝐴) = (𝐹𝐴)
2 funopfvb 6206 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = (𝐹𝐴) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
31, 2mpbii 223 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4161  dom cdm 5084  Fun wfun 5851  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  funfvbrb  6296  fvimacnv  6298  fnopfv  6317  fvelrn  6318  dff3  6338  fnsnb  6397  funfvima3  6460  wfrlem17  7391  tfrlem9a  7442  fundmen  7990  adj1  28680  fgreu  29355  bnj145OLD  30556
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