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Theorem funbrfv 6129
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 5807 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5071 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 486 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 4581 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦𝐴𝐹𝐵))
54anbi2d 735 . . . . 5 (𝑦 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑦) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2620 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝐴) = 𝑦 ↔ (𝐹𝐴) = 𝐵))
75, 6imbi12d 332 . . . 4 (𝑦 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)))
8 funeu 5814 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9 tz6.12-1 6105 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
108, 9sylan2 489 . . . . 5 ((𝐴𝐹𝑦 ∧ (Fun 𝐹𝐴𝐹𝑦)) → (𝐹𝐴) = 𝑦)
1110anabss7 857 . . . 4 ((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
127, 11vtoclg 3238 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵))
133, 12mpcom 37 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)
1413ex 448 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  ∃!weu 2457  Vcvv 3172   class class class wbr 4577  Rel wrel 5033  Fun wfun 5784  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  funopfv  6130  fnbrfvb  6131  fvelima  6143  fvi  6150  opabiota  6156  fmptco  6288  fliftfun  6440  fliftval  6444  tfrlem5  7340  fpwwe2  9321  nqerid  9611  sum0  14245  sumz  14246  fsumsers  14252  isumclim  14276  ntrivcvgfvn0  14416  ntrivcvgtail  14417  zprodn0  14454  iprodclim  14514  idinv  16218  cnextfvval  21621  cnextfres  21625  dvadd  23426  dvmul  23427  dvco  23433  dvcj  23436  dvrec  23441  dvcnv  23461  dvef  23464  ftc1cn  23527  ulmdv  23878  minvecolem4b  26924  minvecolem4  26926  hlimuni  27285  chscllem4  27689  fmptcof2  28645  fvtransport  31115  fvray  31224  fvline  31227  ftc1cnnc  32457  frege124d  36875
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