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Theorem fnbrfvb 6274
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 2651 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 fvex 6239 . . . . 5 (𝐹𝐵) ∈ V
3 eqeq2 2662 . . . . . . 7 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
4 breq2 4689 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹𝐵)))
53, 4bibi12d 334 . . . . . 6 (𝑥 = (𝐹𝐵) → (((𝐹𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
65imbi2d 329 . . . . 5 (𝑥 = (𝐹𝐵) → (((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))))
7 fneu 6033 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8 tz6.12c 6251 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
97, 8syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
102, 6, 9vtocl 3290 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
111, 10mpbii 223 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
12 breq2 4689 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1311, 12syl5ibcom 235 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
14 fnfun 6026 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
15 funbrfv 6272 . . . 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 202 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  ∃!weu 2498   class class class wbr 4685  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fnopfvb  6275  funbrfvb  6276  fnbrfvb2  6278  dffn5  6280  feqmptdf  6290  fnsnfv  6297  fndmdif  6361  dffo4  6415  dff13  6552  isomin  6627  isoini  6628  br1steqg  7232  br2ndeqg  7233  1stconst  7310  2ndconst  7311  fsplit  7327  seqomlem3  7592  seqomlem4  7593  nqerrel  9792  imasleval  16248  znleval  19951  axcontlem5  25893  elnlfn  28915  adjbd1o  29072  fcoinvbr  29545  elintfv  31788  fv1stcnv  31804  fv2ndcnv  31805  trpredpred  31852  scutun12  32042  madeval2  32061  fvbigcup  32134  fvsingle  32152  imageval  32162  brfullfun  32180  bj-mptval  33195  unccur  33522  poimirlem2  33541  poimirlem23  33562  pw2f1ocnv  37921  brcoffn  38645  funressnfv  41529  fnbrafvb  41555
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