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Theorem funbreq 35273
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 35272 . . 3 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
54expdimp 452 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
6 breq2 5145 . . . 4 (𝐵 = 𝐶 → (𝐴𝐹𝐵𝐴𝐹𝐶))
76biimpcd 248 . . 3 (𝐴𝐹𝐵 → (𝐵 = 𝐶𝐴𝐹𝐶))
87adantl 481 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐵 = 𝐶𝐴𝐹𝐶))
95, 8impbid 211 1 ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468   class class class wbr 5141  Fun wfun 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6538
This theorem is referenced by: (None)
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