ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnbr GIF version

Theorem fnbr 5129
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 5125 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
2 releldm 4683 . . 3 ((Rel 𝐹𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
31, 2sylan 278 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4 fndm 5126 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54eleq2d 2158 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
65biimpa 291 . 2 ((𝐹 Fn 𝐴𝐵 ∈ dom 𝐹) → 𝐵𝐴)
73, 6syldan 277 1 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1439   class class class wbr 3851  dom cdm 4452  Rel wrel 4457   Fn wfn 5023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-dm 4462  df-fun 5030  df-fn 5031
This theorem is referenced by:  fnop  5130  dffn5im  5363  dffo4  5461  dffo5  5462  tfrlem5  6093
  Copyright terms: Public domain W3C validator