Theorem eufnfv 5884

Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Hypotheses

Ref	Expression
eufnfv.1	⊢ 𝐴 ∈ V
eufnfv.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eufnfv	⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eufnfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufnfv.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	mptex 5879	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
3		eqeq2 2241	. . . . . 6 ⊢ (𝑦 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 = 𝑦 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	3	bibi2d 232	. . . . 5 ⊢ (𝑦 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))))
5	4	albidv 1872	. . . 4 ⊢ (𝑦 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))))
6	2, 5	spcev 2901	. . 3 ⊢ (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑦∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑦))
7		eufnfv.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8		eqid 2231	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	fnmpti 5461	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
10		fneq1 5418	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
11	9, 10	mpbiri 168	. . . . 5 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑓 Fn 𝐴)
12	11	pm4.71ri 392	. . . 4 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
13		dffn5im 5691	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
14	13	eqeq1d 2240	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)))
15		funfvex 5656	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ V)
16	15	funfni 5432	. . . . . . . 8 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ V)
17	16	ralrimiva 2605	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V)
18		mpteqb 5737	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
19	17, 18	syl 14	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
20	14, 19	bitrd 188	. . . . 5 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
21	20	pm5.32i 454	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
22	12, 21	bitr2i 185	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))
23	6, 22	mpg 1499	. 2 ⊢ ∃𝑦∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑦)
24		df-eu 2082	. 2 ⊢ (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ ∃𝑦∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑦))
25	23, 24	mpbir 146	1 ⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ↦ cmpt 4150   Fn wfn 5321  ‘cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by: (None)