Theorem fununiq 36282

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 6553	. 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 5119	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
6	5	3adant3 1150	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝐵))
7		breq12 5119	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
8	7	3adant2 1149	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝐹𝑧 ↔ 𝐴𝐹𝐶))
9	6, 8	anbi12d 644	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶)))
10		eqeq12 2783	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
11	10	3adant1 1148	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 = 𝑧 ↔ 𝐵 = 𝐶))
12	9, 11	imbi12d 347	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
13	12	spc3gv 3566	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)))
14	2, 3, 4, 13	mp3an 1490	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))
15	1, 14	simplbiim 514	1 ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  Vcvv 3458   class class class wbr 5114  Rel wrel 5671  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-fun 6545

This theorem is used by:  funbreq  36283