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Theorem fnop 5982
Description: The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
fnop ((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)

Proof of Theorem fnop
StepHypRef Expression
1 df-br 4645 . 2 (𝐵𝐹𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹)
2 fnbr 5981 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
31, 2sylan2br 493 1 ((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  cop 4174   class class class wbr 4644   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-dm 5114  df-fun 5878  df-fn 5879
This theorem is referenced by:  2elresin  5990  wfrlem12  7411  tfrlem9  7466  wlkp1lem2  26552  frrlem11  31766  poimirlem4  33384
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