Theorem fnbr 6589

Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Assertion

Ref	Expression
fnbr	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Proof of Theorem fnbr

Step	Hyp	Ref	Expression
1		fnrel 6583	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		releldm 5883	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
3	1, 2	sylan 580	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4		fndm 6584	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	eleq2d 2817	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
6	5	biimpa 476	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴)
7	3, 6	syldan 591	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111   class class class wbr 5089  dom cdm 5614  Rel wrel 5619   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624  df-fun 6483  df-fn 6484

This theorem is referenced by:  fnop  6590  fvelima2  6874  dffn5  6880  feqmptdf  6892  dffo4  7036  dffo5  7037  tfrlem5  8299  occllem  31283  chscllem2  31618  tfsconcat0i  43448  brcoffn  44133  dfafn5a  47270