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Theorem funbrafv2b 45286
Description: Function value in terms of a binary relation, analogous to funbrfv2b 6898. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
funbrafv2b (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Proof of Theorem funbrafv2b
StepHypRef Expression
1 funrel 6516 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5898 . . . . 5 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
32ex 414 . . . 4 (Rel 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
41, 3syl 17 . . 3 (Fun 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
54pm4.71rd 564 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
6 funbrafvb 45283 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))
76pm5.32da 580 . 2 (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
85, 7bitr4d 282 1 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5104  dom cdm 5632  Rel wrel 5637  Fun wfun 6488  '''cafv 45244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6446  df-fun 6496  df-fn 6497  df-fv 6502  df-aiota 45212  df-dfat 45246  df-afv 45247
This theorem is referenced by: (None)
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