Theorem eufnfv 7163

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 7157	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
3		eqeq2 2743	. . . . . 6 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 = 𝑧 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	3	bibi2d 342	. . . . 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 1921	. . . 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 3556	. . 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 2731	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	fnmpti 6624	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
10		fneq1 6572	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
11	9, 10	mpbiri 258	. . . . 5 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑓 Fn 𝐴)
12	11	pm4.71ri 560	. . . 4 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))
13		dffn5 6880	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
14		eqeq1 2735	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)))
15	13, 14	sylbi 217	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)))
16		fvex 6835	. . . . . . . 8 ⊢ (𝑓‘𝑥) ∈ V
17	16	rgenw 3051	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ V
18		mpteqb 6948	. . . . . . 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 287	. . . . 5 ⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
21	20	pm5.32i 574	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵))
22	12, 21	bitr2i 276	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))
23	6, 22	mpg 1798	. 2 ⊢ ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧)
24		eu6 2569	. 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 231	1 ⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  Vcvv 3436   ↦ cmpt 5170   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by: (None)