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Theorem fnop 6143
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 4793 . 2 (𝐵𝐹𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹)
2 fnbr 6142 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
31, 2sylan2br 494 1 ((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2127  cop 4315   class class class wbr 4792   Fn wfn 6032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-dm 5264  df-fun 6039  df-fn 6040
This theorem is referenced by:  2elresin  6151  wfrlem12  7583  tfrlem9  7638  wlkp1lem2  26752  frrlem11  32069  poimirlem4  33695
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