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Theorem fnbr 6656
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 6650 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
2 releldm 5940 . . 3 ((Rel 𝐹𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
31, 2sylan 579 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4 fndm 6651 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54eleq2d 2815 . . 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 2099   class class class wbr 5142  dom cdm 5672  Rel wrel 5677   Fn wfn 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-dm 5682  df-fun 6544  df-fn 6545
This theorem is referenced by:  fnop  6657  dffn5  6951  feqmptdf  6963  dffo4  7107  dffo5  7108  tfrlem5  8394  occllem  31106  chscllem2  31441  tfsconcat0i  42768  brcoffn  43454  fvelima2  44630  dfafn5a  46534
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