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Theorem fnbr 5029
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 5025 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
2 releldm 4597 . . 3 ((Rel 𝐹𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
31, 2sylan 271 . 2 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4 fndm 5026 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54eleq2d 2123 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
65biimpa 284 . 2 ((𝐹 Fn 𝐴𝐵 ∈ dom 𝐹) → 𝐵𝐴)
73, 6syldan 270 1 ((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409   class class class wbr 3792  dom cdm 4373  Rel wrel 4378   Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-dm 4383  df-fun 4932  df-fn 4933
This theorem is referenced by:  fnop  5030  dffn5im  5247  dffo4  5343  dffo5  5344  tfrlem5  5961
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