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Theorem funbrfv 6957
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 6583 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5739 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 580 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 5147 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦𝐴𝐹𝐵))
54anbi2d 630 . . . . 5 (𝑦 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑦) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2749 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝐴) = 𝑦 ↔ (𝐹𝐴) = 𝐵))
75, 6imbi12d 344 . . . 4 (𝑦 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)))
8 funeu 6591 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9 tz6.12-1 6929 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
108, 9sylan2 593 . . . . 5 ((𝐴𝐹𝑦 ∧ (Fun 𝐹𝐴𝐹𝑦)) → (𝐹𝐴) = 𝑦)
1110anabss7 673 . . . 4 ((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
127, 11vtoclg 3554 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)
1413ex 412 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  Vcvv 3480   class class class wbr 5143  Rel wrel 5690  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  funopfv  6958  fnbrfvb  6959  fvelima2  6961  fvelima  6974  fvelimad  6976  fvi  6985  opabiota  6991  fmptco  7149  fliftfun  7332  fliftval  7336  tfrlem5  8420  fpwwe2  10683  nqerid  10973  sum0  15757  sumz  15758  fsumsers  15764  isumclim  15793  ntrivcvgfvn0  15935  ntrivcvgtail  15936  zprodn0  15975  iprodclim  16034  idinv  17833  cnextfvval  24073  cnextfres  24077  dvadd  25977  dvmul  25978  dvco  25985  dvcj  25988  dvrec  25993  dvcnv  26015  dvef  26018  ftc1cn  26084  ulmdv  26446  minvecolem4b  30897  minvecolem4  30899  hlimuni  31257  chscllem4  31659  fmptcof2  32667  fvtransport  36033  fvray  36142  fvline  36145  ftc1cnnc  37699  iscard4  43546  frege124d  43774
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