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Theorem funbrfv2b 6150
 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 5821 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5279 . . . . 5 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
32ex 449 . . . 4 (Rel 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
41, 3syl 17 . . 3 (Fun 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
54pm4.71rd 665 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
6 funbrfvb 6148 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
76pm5.32da 671 . 2 (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
85, 7bitr4d 270 1 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804 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-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  brtpos2  7245  mpt2curryd  7282  xpcomco  7935  fseqenlem2  8731  fpwwe2  9344  joinfval  16824  joinfval2  16825  meetfval  16838  meetfval2  16839  tayl0  23920  ofpreima  28848  funcnvmptOLD  28850  funcnvmpt  28851  curf  32557  uncf  32558  curunc  32561  fperdvper  38808
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