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Theorem fununiq 31363
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 5860 . 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 4623 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦𝐴𝐹𝐵))
653adant3 1079 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝐹𝑦𝐴𝐹𝐵))
7 breq12 4623 . . . . . . . 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 2639 . . . . . . 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 3289 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶)))
142, 3, 4, 13mp3an 1421 . . 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 1478   = wceq 1480  wcel 1992  Vcvv 3191   class class class wbr 4618  Rel wrel 5084  Fun wfun 5844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-cnv 5087  df-co 5088  df-fun 5852
This theorem is referenced by:  funbreq  31364
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