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Theorem fnbrfvb 6944
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 2731 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 fvex 6904 . . . . 5 (𝐹𝐵) ∈ V
3 eqeq2 2743 . . . . . . 7 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
4 breq2 5152 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹𝐵)))
53, 4bibi12d 345 . . . . . 6 (𝑥 = (𝐹𝐵) → (((𝐹𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
65imbi2d 340 . . . . 5 (𝑥 = (𝐹𝐵) → (((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))))
7 fneu 6659 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8 tz6.12c 6913 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
97, 8syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
102, 6, 9vtocl 3545 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
111, 10mpbii 232 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
12 breq2 5152 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1311, 12syl5ibcom 244 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
14 fnfun 6649 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
15 funbrfv 6942 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1614, 15syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1716adantr 480 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1813, 17impbid 211 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  ∃!weu 2561   class class class wbr 5148  Fun wfun 6537   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  fnopfvb  6945  funbrfvb  6946  fnbrfvb2  6948  dffn5  6950  feqmptdf  6962  fnsnfv  6970  fnsnfvOLD  6971  fndmdif  7043  dffo4  7104  dff13  7257  isomin  7337  isoini  7338  br1steqg  8001  br2ndeqg  8002  1stconst  8091  2ndconst  8092  fsplit  8108  seqomlem3  8458  seqomlem4  8459  nqerrel  10933  imasleval  17494  znleval  21421  scutun12  27658  madeval2  27695  axcontlem5  28661  elnlfn  31616  adjbd1o  31773  fcoinvbr  32271  fv1stcnv  35220  fv2ndcnv  35221  fvbigcup  35346  fvsingle  35364  imageval  35374  brfullfun  35392  bj-mptval  36465  unccur  36938  poimirlem2  36957  poimirlem23  36978  pw2f1ocnv  42242  tfsconcat0i  42561  tfsconcatrev  42564  brcoffn  43247  funressnfv  46215  fnbrafvb  46324
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