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Theorem fnbrfvb 6131
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 2609 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 fvex 6098 . . . . 5 (𝐹𝐵) ∈ V
3 eqeq2 2620 . . . . . . 7 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
4 breq2 4581 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹𝐵)))
53, 4bibi12d 333 . . . . . 6 (𝑥 = (𝐹𝐵) → (((𝐹𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
65imbi2d 328 . . . . 5 (𝑥 = (𝐹𝐵) → (((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))))
7 fneu 5895 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8 tz6.12c 6108 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
97, 8syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
102, 6, 9vtocl 3231 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
111, 10mpbii 221 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
12 breq2 4581 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1311, 12syl5ibcom 233 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
14 fnfun 5888 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
15 funbrfv 6129 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1614, 15syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1716adantr 479 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1813, 17impbid 200 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  ∃!weu 2457   class class class wbr 4577  Fun wfun 5784   Fn wfn 5785  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 2033  ax-13 2233  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-fn 5793  df-fv 5798
This theorem is referenced by:  fnopfvb  6132  funbrfvb  6133  dffn5  6136  feqmptdf  6146  fnsnfv  6153  fndmdif  6214  dffo4  6268  dff13  6394  isomin  6465  isoini  6466  1stconst  7129  2ndconst  7130  fsplit  7146  seqomlem3  7411  seqomlem4  7412  nqerrel  9610  imasleval  15970  znleval  19667  axcontlem5  25566  elnlfn  27977  adjbd1o  28134  fcoinvbr  28605  br1steq  30723  br2ndeq  30724  fv1stcnv  30731  fv2ndcnv  30732  trpredpred  30778  fvbigcup  30985  fvsingle  31003  imageval  31013  brfullfun  31031  bj-mptval  32047  unccur  32358  poimirlem2  32377  poimirlem23  32398  pw2f1ocnv  36418  brcoffn  37144  funressnfv  39654  fnbrafvb  39681
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