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Theorem fnbrfvb 6693
 Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fnbrfvb ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))

Proof of Theorem fnbrfvb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 fvex 6658 . . . . 5 (𝐹𝐵) ∈ V
3 eqeq2 2810 . . . . . . 7 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
4 breq2 5034 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹𝐵)))
53, 4bibi12d 349 . . . . . 6 (𝑥 = (𝐹𝐵) → (((𝐹𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
65imbi2d 344 . . . . 5 (𝑥 = (𝐹𝐵) → (((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))))
7 fneu 6432 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8 tz6.12c 6670 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
97, 8syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
102, 6, 9vtocl 3507 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
111, 10mpbii 236 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
12 breq2 5034 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1311, 12syl5ibcom 248 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
14 fnfun 6423 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
15 funbrfv 6691 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1614, 15syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1716adantr 484 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1813, 17impbid 215 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!weu 2628   class class class wbr 5030  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332 This theorem is referenced by:  fnopfvb  6694  funbrfvb  6695  fnbrfvb2  6697  dffn5  6699  feqmptdf  6710  fnsnfv  6718  fndmdif  6789  dffo4  6846  dff13  6991  isomin  7069  isoini  7070  br1steqg  7695  br2ndeqg  7696  1stconst  7780  2ndconst  7781  fsplit  7797  fsplitOLD  7798  seqomlem3  8073  seqomlem4  8074  nqerrel  10345  imasleval  16808  znleval  20250  axcontlem5  26769  elnlfn  29718  adjbd1o  29875  fcoinvbr  30378  fv1stcnv  33145  fv2ndcnv  33146  trpredpred  33192  scutun12  33396  madeval2  33415  fvbigcup  33488  fvsingle  33506  imageval  33516  brfullfun  33534  bj-mptval  34548  unccur  35056  poimirlem2  35075  poimirlem23  35096  pw2f1ocnv  39993  brcoffn  40748  funressnfv  43650  fnbrafvb  43725
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