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Theorem funbrfv 6144
 Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
funbrfv (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))

Proof of Theorem funbrfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funrel 5821 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5081 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 487 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 4587 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦𝐴𝐹𝐵))
54anbi2d 736 . . . . 5 (𝑦 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑦) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2621 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝐴) = 𝑦 ↔ (𝐹𝐴) = 𝐵))
75, 6imbi12d 333 . . . 4 (𝑦 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)))
8 funeu 5828 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9 tz6.12-1 6120 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
108, 9sylan2 490 . . . . 5 ((𝐴𝐹𝑦 ∧ (Fun 𝐹𝐴𝐹𝑦)) → (𝐹𝐴) = 𝑦)
1110anabss7 858 . . . 4 ((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
127, 11vtoclg 3239 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵))
133, 12mpcom 37 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)
1413ex 449 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173   class class class wbr 4583  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-fv 5812 This theorem is referenced by:  funopfv  6145  fnbrfvb  6146  fvelima  6158  fvi  6165  opabiota  6171  fmptco  6303  fliftfun  6462  fliftval  6466  tfrlem5  7363  fpwwe2  9344  nqerid  9634  sum0  14299  sumz  14300  fsumsers  14306  isumclim  14330  ntrivcvgfvn0  14470  ntrivcvgtail  14471  zprodn0  14508  iprodclim  14568  idinv  16272  cnextfvval  21679  cnextfres  21683  dvadd  23509  dvmul  23510  dvco  23516  dvcj  23519  dvrec  23524  dvcnv  23544  dvef  23547  ftc1cn  23610  ulmdv  23961  minvecolem4b  27118  minvecolem4  27120  hlimuni  27479  chscllem4  27883  fmptcof2  28839  fvtransport  31309  fvray  31418  fvline  31421  ftc1cnnc  32654  frege124d  37072
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