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Theorem funbreq 35751
Description: An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
funbreq.1 𝐴 ∈ V
funbreq.2 𝐵 ∈ V
funbreq.3 𝐶 ∈ V
Assertion
Ref Expression
funbreq ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))

Proof of Theorem funbreq
StepHypRef Expression
1 funbreq.1 . . . 4 𝐴 ∈ V
2 funbreq.2 . . . 4 𝐵 ∈ V
3 funbreq.3 . . . 4 𝐶 ∈ V
41, 2, 3fununiq 35750 . . 3 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
54expdimp 452 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
6 breq2 5152 . . . 4 (𝐵 = 𝐶 → (𝐴𝐹𝐵𝐴𝐹𝐶))
76biimpcd 249 . . 3 (𝐴𝐹𝐵 → (𝐵 = 𝐶𝐴𝐹𝐶))
87adantl 481 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐵 = 𝐶𝐴𝐹𝐶))
95, 8impbid 212 1 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   class class class wbr 5148  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by: (None)
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