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Theorem fununiq 31793
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 5936 . 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 4690 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦𝐴𝐹𝐵))
653adant3 1101 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑦𝐴𝐹𝐵))
7 breq12 4690 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
873adant2 1100 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑧𝐴𝐹𝐶))
96, 8anbi12d 747 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵𝐴𝐹𝐶)))
10 eqeq12 2664 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
11103adant1 1099 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦 = 𝑧𝐵 = 𝐶))
129, 11imbi12d 333 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
1312spc3gv 3329 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
142, 3, 4, 13mp3an 1464 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))
1514adantl 481 . 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 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685  Rel wrel 5148  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-cnv 5151  df-co 5152  df-fun 5928
This theorem is referenced by:  funbreq  31794
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