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Theorem fununiq 35769
Description: The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
fununiq.1 𝐴 ∈ V
fununiq.2 𝐵 ∈ V
fununiq.3 𝐶 ∈ V
Assertion
Ref Expression
fununiq (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))

Proof of Theorem fununiq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 6571 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2 fununiq.1 . . 3 𝐴 ∈ V
3 fununiq.2 . . 3 𝐵 ∈ V
4 fununiq.3 . . 3 𝐶 ∈ V
5 breq12 5148 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦𝐴𝐹𝐵))
653adant3 1133 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑦𝐴𝐹𝐵))
7 breq12 5148 . . . . . . 7 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
873adant2 1132 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
96, 8anbi12d 632 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵𝐴𝐹𝐶)))
10 eqeq12 2754 . . . . . 6 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
11103adant1 1131 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
129, 11imbi12d 344 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
1312spc3gv 3604 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
142, 3, 4, 13mp3an 1463 . 2 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
151, 14simplbiim 504 1 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480   class class class wbr 5143  Rel wrel 5690  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563
This theorem is referenced by:  funbreq  35770
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