MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnbrfvb Structured version   Visualization version   GIF version

Theorem fnbrfvb 6895
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 2736 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 fvex 6855 . . . . 5 (𝐹𝐵) ∈ V
3 eqeq2 2748 . . . . . . 7 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
4 breq2 5109 . . . . . . 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 6612 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8 tz6.12c 6864 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
97, 8syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
102, 6, 9vtocl 3518 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
111, 10mpbii 232 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
12 breq2 5109 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1311, 12syl5ibcom 244 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
14 fnfun 6602 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
15 funbrfv 6893 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1614, 15syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1716adantr 481 . 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 396   = wceq 1541  wcel 2106  ∃!weu 2566   class class class wbr 5105  Fun wfun 6490   Fn wfn 6491  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  fnopfvb  6896  funbrfvb  6897  fnbrfvb2  6899  dffn5  6901  feqmptdf  6912  fnsnfv  6920  fnsnfvOLD  6921  fndmdif  6992  dffo4  7053  dff13  7202  isomin  7282  isoini  7283  br1steqg  7943  br2ndeqg  7944  1stconst  8032  2ndconst  8033  fsplit  8049  seqomlem3  8398  seqomlem4  8399  nqerrel  10868  imasleval  17423  znleval  20961  scutun12  27149  madeval2  27183  axcontlem5  27917  elnlfn  30870  adjbd1o  31027  fcoinvbr  31526  fv1stcnv  34351  fv2ndcnv  34352  fvbigcup  34487  fvsingle  34505  imageval  34515  brfullfun  34533  bj-mptval  35588  unccur  36061  poimirlem2  36080  poimirlem23  36101  pw2f1ocnv  41347  brcoffn  42292  funressnfv  45267  fnbrafvb  45376
  Copyright terms: Public domain W3C validator