Theorem fununiq 35044

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 6552	. 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 5152	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
6	5	3adant3 1130	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
7		breq12 5152	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
8	7	3adant2 1129	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
9	6, 8	anbi12d 629	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶)))
10		eqeq12 2747	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
11	10	3adant1 1128	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
12	9, 11	imbi12d 343	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
13	12	spc3gv 3593	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
14	2, 3, 4, 13	mp3an 1459	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))
15	1, 14	simplbiim 503	1 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2104  Vcvv 3472   class class class wbr 5147  Rel wrel 5680  Fun wfun 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-fun 6544

This theorem is referenced by:  funbreq  35045