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Theorem funbreq 32182
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 32181 . . 3 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
54expdimp 445 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
6 breq2 4847 . . . 4 (𝐵 = 𝐶 → (𝐴𝐹𝐵𝐴𝐹𝐶))
76biimpcd 241 . . 3 (𝐴𝐹𝐵 → (𝐵 = 𝐶𝐴𝐹𝐶))
87adantl 474 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐵 = 𝐶𝐴𝐹𝐶))
95, 8impbid 204 1 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385   class class class wbr 4843  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-cnv 5320  df-co 5321  df-fun 6103
This theorem is referenced by: (None)
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