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Theorem fnbr 6648
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 6642 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
2 releldm 5934 . . 3 ((Rel 𝐹𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
31, 2sylan 579 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4 fndm 6643 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54eleq2d 2811 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
65biimpa 476 . 2 ((𝐹 Fn 𝐴𝐵 ∈ dom 𝐹) → 𝐵𝐴)
73, 6syldan 590 1 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098   class class class wbr 5139  dom cdm 5667  Rel wrel 5672   Fn wfn 6529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-dm 5677  df-fun 6536  df-fn 6537
This theorem is referenced by:  fnop  6649  dffn5  6941  feqmptdf  6953  dffo4  7095  dffo5  7096  tfrlem5  8376  occllem  31028  chscllem2  31363  tfsconcat0i  42609  brcoffn  43295  fvelima2  44474  dfafn5a  46378
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