Theorem fnimage 36514

Description: Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fnimage	⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}

Distinct variable group:   𝑥,𝑅

Proof of Theorem fnimage

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimage 36513	. 2 ⊢ Fun Image𝑅
2		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3457	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	brimage 36511	. . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦))
5		eqvisset 3473	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V)
6	4, 5	sylbi 220	. . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
7	6	exlimiv 1963	. . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
8		eqid 2762	. . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)
9		brimageg 36512	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)))
10	2, 9	mpan 703	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)))
11	8, 10	mpbiri 261	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦Image𝑅(𝑅 “ 𝑦))
12		breq2 5111	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦)))
13	12	spcegv 3554	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
14	11, 13	mpd 16	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
15	7, 14	impbii 212	. . . 4 ⊢ (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅 “ 𝑦) ∈ V)
16	2	eldm 5888	. . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17		imaeq2 6056	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2847	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	2, 18	elab 3636	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	15, 16, 19	3bitr4i 306	. . 3 ⊢ (𝑦 ∈ dom Image𝑅 ↔ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
21	20	eqriv 2759	. 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
22		df-fn 6540	. 2 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}))
23	1, 21, 22	mpbir2an 724	1 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  Vcvv 3453   class class class wbr 5107  dom cdm 5659   “ cima 5662  Fun wfun 6531   Fn wfn 6532  Imagecimage 36425

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4202  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7990  df-2nd 7991  df-txp 36439  df-image 36449

This theorem is used by:  imageval  36515