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Theorem funbrfv 5453
 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 5135 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 4575 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 281 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 3928 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦𝐴𝐹𝐵))
54anbi2d 459 . . . . 5 (𝑦 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑦) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2147 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝐴) = 𝑦 ↔ (𝐹𝐴) = 𝐵))
75, 6imbi12d 233 . . . 4 (𝑦 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)))
8 funeu 5143 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9 tz6.12-1 5441 . . . . . 6 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
108, 9sylan2 284 . . . . 5 ((𝐴𝐹𝑦 ∧ (Fun 𝐹𝐴𝐹𝑦)) → (𝐹𝐴) = 𝑦)
1110anabss7 572 . . . 4 ((Fun 𝐹𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
127, 11vtoclg 2741 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵))
133, 12mpcom 36 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹𝐴) = 𝐵)
1413ex 114 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∃!weu 1997  Vcvv 2681   class class class wbr 3924  Rel wrel 4539  Fun wfun 5112  ‘cfv 5118 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126 This theorem is referenced by:  funopfv  5454  fnbrfvb  5455  fvelima  5466  fvi  5471  fmptco  5579  fliftfun  5690  fliftval  5694  tfrlem5  6204  sum0  11150  isumz  11151  fsumsersdc  11157  isumclim  11183  dvaddxx  12825  dvmulxx  12826  dvcj  12831  dvrecap  12835  dvef  12845  pilem3  12853
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