Theorem fnop 6678

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) → 𝐵 ∈ 𝐴)

Proof of Theorem fnop

Step	Hyp	Ref	Expression
1		df-br 5109	. 2 ⊢ (𝐵𝐹𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹)
2		fnbr 6677	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)
3	1, 2	sylan2br 607	1 ⊢ ((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ⟨cop 4840   class class class wbr 5108   Fn wfn 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-fun 6568  df-fn 6569

This theorem is used by:  2elresin  6690  tfrlem9  8431  wlkp1lem2  30310  poimirlem4  38810