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Theorem fnbr 4944
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
StepHypRef Expression
1 fnrel 4940 . . 3 (𝐹 Fn A → Rel 𝐹)
2 releldm 4512 . . 3 ((Rel 𝐹 B𝐹𝐶) → B dom 𝐹)
31, 2sylan 267 . 2 ((𝐹 Fn A B𝐹𝐶) → B dom 𝐹)
4 fndm 4941 . . . 4 (𝐹 Fn A → dom 𝐹 = A)
54eleq2d 2104 . . 3 (𝐹 Fn A → (B dom 𝐹B A))
65biimpa 280 . 2 ((𝐹 Fn A B dom 𝐹) → B A)
73, 6syldan 266 1 ((𝐹 Fn A B𝐹𝐶) → B A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390   class class class wbr 3755  dom cdm 4288  Rel wrel 4293   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  fnop  4945  dffn5im  5162  dffo4  5258  dffo5  5259  tfrlem5  5871
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