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Theorem funbrfv2b 5434
Description: Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
Assertion
Ref Expression
funbrfv2b (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))

Proof of Theorem funbrfv2b
StepHypRef Expression
1 funrel 5110 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 4744 . . . . 5 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
32ex 114 . . . 4 (Rel 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
41, 3syl 14 . . 3 (Fun 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
54pm4.71rd 391 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
6 funbrfvb 5432 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
76pm5.32da 447 . 2 (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
85, 7bitr4d 190 1 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465   class class class wbr 3899  dom cdm 4509  Rel wrel 4514  Fun wfun 5087  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  brtpos2  6116  xpcomco  6688
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