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Theorem fnbrfvb 5468
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 2140 . . . 4 (𝐹𝐵) = (𝐹𝐵)
2 funfvex 5444 . . . . . 6 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐹𝐵) ∈ V)
32funfni 5229 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ V)
4 eqeq2 2150 . . . . . . . 8 (𝑥 = (𝐹𝐵) → ((𝐹𝐵) = 𝑥 ↔ (𝐹𝐵) = (𝐹𝐵)))
5 breq2 3939 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹𝐵)))
64, 5bibi12d 234 . . . . . . 7 (𝑥 = (𝐹𝐵) → (((𝐹𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
76imbi2d 229 . . . . . 6 (𝑥 = (𝐹𝐵) → (((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))))
8 fneu 5233 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
9 tz6.12c 5457 . . . . . . 7 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
108, 9syl 14 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑥𝐵𝐹𝑥))
117, 10vtoclg 2749 . . . . 5 ((𝐹𝐵) ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵))))
123, 11mpcom 36 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = (𝐹𝐵) ↔ 𝐵𝐹(𝐹𝐵)))
131, 12mpbii 147 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹𝐵))
14 breq2 3939 . . 3 ((𝐹𝐵) = 𝐶 → (𝐵𝐹(𝐹𝐵) ↔ 𝐵𝐹𝐶))
1513, 14syl5ibcom 154 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
16 fnfun 5226 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
17 funbrfv 5466 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1816, 17syl 14 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
1918adantr 274 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹𝐵) = 𝐶))
2015, 19impbid 128 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  ∃!weu 2000  Vcvv 2689   class class class wbr 3935  Fun wfun 5123   Fn wfn 5124  cfv 5129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-iota 5094  df-fun 5131  df-fn 5132  df-fv 5137
This theorem is referenced by:  fnopfvb  5469  funbrfvb  5470  dffn5im  5473  fnsnfv  5486  fndmdif  5531  dffo4  5574  dff13  5675  isoini  5725  1stconst  6124  2ndconst  6125  pw1nct  13364
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