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Theorem fnbr 6676
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 𝐴𝐵𝐹𝐶) → 𝐵𝐴)

Proof of Theorem fnbr
StepHypRef Expression
1 fnrel 6670 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
2 releldm 5955 . . 3 ((Rel 𝐹𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
31, 2sylan 580 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4 fndm 6671 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54eleq2d 2827 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
65biimpa 476 . 2 ((𝐹 Fn 𝐴𝐵 ∈ dom 𝐹) → 𝐵𝐴)
73, 6syldan 591 1 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108   class class class wbr 5143  dom cdm 5685  Rel wrel 5690   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  fnop  6677  fvelima2  6961  dffn5  6967  feqmptdf  6979  dffo4  7123  dffo5  7124  tfrlem5  8420  occllem  31322  chscllem2  31657  tfsconcat0i  43358  brcoffn  44043  dfafn5a  47172
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