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Theorem funbreq 30914
 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 30913 . . 3 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
54expdimp 452 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
6 breq2 4587 . . . 4 (𝐵 = 𝐶 → (𝐴𝐹𝐵𝐴𝐹𝐶))
76biimpcd 238 . . 3 (𝐴𝐹𝐵 → (𝐵 = 𝐶𝐴𝐹𝐶))
87adantl 481 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐵 = 𝐶𝐴𝐹𝐶))
95, 8impbid 201 1 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by: (None)
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