Theorem eufnfv 5895

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 5890	. . . 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 2902	. . 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 5468	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
10		fneq1 5425	. . . . . 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 5700	. . . . . . 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 5665	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ V)
16	15	funfni 5439	. . . . . . . 8 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ V)
17	16	ralrimiva 2606	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V)
18		mpteqb 5746	. . . . . . 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 1500	. 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 1396   = wceq 1398  ∃wex 1541  ∃!weu 2079   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ↦ cmpt 4155   Fn wfn 5328  ‘cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341

This theorem is referenced by: (None)