Theorem fnbr 6541

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 6535	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		releldm 5853	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
3	1, 2	sylan 580	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4		fndm 6536	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	eleq2d 2824	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
6	5	biimpa 477	. 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 396   ∈ wcel 2106   class class class wbr 5074  dom cdm 5589  Rel wrel 5594   Fn wfn 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-fun 6435  df-fn 6436

This theorem is referenced by:  fnop  6542  dffn5  6828  feqmptdf  6839  dffo4  6979  dffo5  6980  tfrlem5  8211  occllem  29665  chscllem2  30000  brcoffn  41640  fvelima2  42806  dfafn5a  44652