Theorem fununiq 35763

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 6524	. 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 5115	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
6	5	3adant3 1132	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
7		breq12 5115	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
8	7	3adant2 1131	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
9	6, 8	anbi12d 632	. . . . 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 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 3573	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
14	2, 3, 4, 13	mp3an 1463	. 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 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3450   class class class wbr 5110  Rel wrel 5646  Fun wfun 6508

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-fun 6516

This theorem is referenced by:  funbreq  35764