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Theorem fununiq 31643
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 5886 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2 fununiq.1 . . . 4 𝐴 ∈ V
3 fununiq.2 . . . 4 𝐵 ∈ V
4 fununiq.3 . . . 4 𝐶 ∈ V
5 breq12 4649 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦𝐴𝐹𝐵))
653adant3 1079 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑦𝐴𝐹𝐵))
7 breq12 4649 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
873adant2 1078 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
96, 8anbi12d 746 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵𝐴𝐹𝐶)))
10 eqeq12 2633 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
11103adant1 1077 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
129, 11imbi12d 334 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
1312spc3gv 3293 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
142, 3, 4, 13mp3an 1422 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
1514adantl 482 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
161, 15sylbi 207 1 (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  Vcvv 3195   class class class wbr 4644  Rel wrel 5109  Fun wfun 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-cnv 5112  df-co 5113  df-fun 5878
This theorem is referenced by:  funbreq  31644
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