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Theorem fununiq 34382
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 6511 . 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 5115 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦𝐴𝐹𝐵))
653adant3 1133 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑦𝐴𝐹𝐵))
7 breq12 5115 . . . . . . 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 345 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
1312spc3gv 3566 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
142, 3, 4, 13mp3an 1462 . 2 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
151, 14simplbiim 506 1 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  Vcvv 3448   class class class wbr 5110  Rel wrel 5643  Fun wfun 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-fun 6503
This theorem is referenced by:  funbreq  34383
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