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Theorem funbreq 32910
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 32909 . . 3 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
54expdimp 453 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
6 breq2 5061 . . . 4 (𝐵 = 𝐶 → (𝐴𝐹𝐵𝐴𝐹𝐶))
76biimpcd 250 . . 3 (𝐴𝐹𝐵 → (𝐵 = 𝐶𝐴𝐹𝐶))
87adantl 482 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐵 = 𝐶𝐴𝐹𝐶))
95, 8impbid 213 1 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   class class class wbr 5057  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-cnv 5556  df-co 5557  df-fun 6350
This theorem is referenced by: (None)
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