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Theorem funbrafv 39887
 Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6144. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
funbrafv (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Proof of Theorem funbrafv
StepHypRef Expression
1 funrel 5821 . . 3 (Fun 𝐹 → Rel 𝐹)
2 releldm 5279 . . . . . . . 8 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3 funbrafvb 39885 . . . . . . . . . 10 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))
43biimprd 237 . . . . . . . . 9 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
54expcom 450 . . . . . . . 8 (𝐴 ∈ dom 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
62, 5syl 17 . . . . . . 7 ((Rel 𝐹𝐴𝐹𝐵) → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
76ex 449 . . . . . 6 (Rel 𝐹 → (𝐴𝐹𝐵 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))))
87com14 94 . . . . 5 (𝐴𝐹𝐵 → (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵))))
98pm2.43i 50 . . . 4 (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵)))
109com13 86 . . 3 (Rel 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
111, 10syl 17 . 2 (Fun 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
1211pm2.43i 50 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  Rel wrel 5043  Fun wfun 5798  '''cafv 39843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-dfat 39845  df-afv 39846 This theorem is referenced by:  afvelima  39896
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