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Theorem funbrfv 6376
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 6049 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5298 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 563 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 4791 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦𝐴𝐹𝐵))
54anbi2d 608 . . . . 5 (𝑦 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑦) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2782 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝐴) = 𝑦 ↔ (𝐹𝐴) = 𝐵))
75, 6imbi12d 333 . . . 4 (𝑦 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)))
8 funeu 6057 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9 tz6.12-1 6352 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
108, 9sylan2 574 . . . . 5 ((𝐴𝐹𝑦 ∧ (Fun 𝐹𝐴𝐹𝑦)) → (𝐹𝐴) = 𝑦)
1110anabss7 646 . . . 4 ((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
127, 11vtoclg 3418 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)
1413ex 397 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  ∃!weu 2618  Vcvv 3351   class class class wbr 4787  Rel wrel 5255  Fun wfun 6026  cfv 6032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040
This theorem is referenced by:  funopfv  6377  fnbrfvb  6378  fvelima  6391  fvi  6398  opabiota  6404  fmptco  6540  fliftfun  6706  fliftval  6710  tfrlem5  7630  fpwwe2  9668  nqerid  9958  sum0  14661  sumz  14662  fsumsers  14668  isumclim  14697  ntrivcvgfvn0  14839  ntrivcvgtail  14840  zprodn0  14877  iprodclim  14936  idinv  16657  cnextfvval  22090  cnextfres  22094  dvadd  23924  dvmul  23925  dvco  23931  dvcj  23934  dvrec  23939  dvcnv  23961  dvef  23964  ftc1cn  24027  ulmdv  24378  minvecolem4b  28075  minvecolem4  28077  hlimuni  28436  chscllem4  28840  fmptcof2  29798  fvtransport  32477  fvray  32586  fvline  32589  ftc1cnnc  33817  frege124d  38580  fvelimad  39977  fvelima2  39994
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