Theorem fununiq 35732

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 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))

Proof of Theorem fununiq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 6583	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2		fununiq.1	. . 3 ⊢ 𝐴 ∈ V
3		fununiq.2	. . 3 ⊢ 𝐵 ∈ V
4		fununiq.3	. . 3 ⊢ 𝐶 ∈ V
5		breq12 5171	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
6	5	3adant3 1132	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
7		breq12 5171	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
8	7	3adant2 1131	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
9	6, 8	anbi12d 631	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶)))
10		eqeq12 2757	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
11	10	3adant1 1130	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
12	9, 11	imbi12d 344	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
13	12	spc3gv 3617	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
14	2, 3, 4, 13	mp3an 1461	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))
15	1, 14	simplbiim 504	1 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  funbreq  35733