Theorem eufnfv 7266

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 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufnfv.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	mptex 7260	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
3		eqeq2 2775	. . . . . 6 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 = 𝑧 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	3	bibi2d 345	. . . . 5 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))))
5	4	albidv 1955	. . . 4 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))))
6	2, 5	spcev 3564	. . 3 ⊢ (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
7		eufnfv.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	fnmpti 6712	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
10		fneq1 6660	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
11	9, 10	mpbiri 261	. . . . 5 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑓 Fn 𝐴)
12	11	pm4.71ri 570	. . . 4 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
13		dffn5 6974	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
14		eqeq1 2767	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)))
15	13, 14	sylbi 220	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)))
16		fvex 6929	. . . . . . . 8 ⊢ (𝑓‘𝑥) ∈ V
17	16	rgenw 3084	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V
18		mpteqb 7044	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
19	17, 18	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)
20	15, 19	bitrdi 290	. . . . 5 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
21	20	pm5.32i 585	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
22	12, 21	bitr2i 279	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))
23	6, 22	mpg 1830	. 2 ⊢ ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧)
24		eu6 2602	. 2 ⊢ (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
25	23, 24	mpbir 234	1 ⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  ∀wral 3080  Vcvv 3454   ↦ cmpt 5192   Fn wfn 6561  ‘cfv 6566

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by: (None)