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Theorem funfvop 6840
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 2739 . 2 (𝐹𝐴) = (𝐹𝐴)
2 funopfvb 6738 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = (𝐹𝐴) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
31, 2mpbii 236 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  cop 4532  dom cdm 5535  Fun wfun 6344  cfv 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6308  df-fun 6352  df-fn 6353  df-fv 6358
This theorem is referenced by:  funfvbrb  6841  fvimacnv  6843  fnopfv  6866  fvelrn  6867  dff3  6889  fnsnb  6951  funfvima3  7022  wfrlem17  8013  tfrlem9a  8064  fundmen  8643  adj1  29881  fgreu  30597  fnsnbt  39831
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