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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnimage | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fnimage | ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimage 32035 | . 2 ⊢ Fun Image𝑅 | |
2 | vex 3203 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3203 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brimage 32033 | . . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦)) |
5 | eqvisset 3211 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V) | |
6 | 4, 5 | sylbi 207 | . . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
7 | 6 | exlimiv 1858 | . . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
8 | eqid 2622 | . . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦) | |
9 | brimageg 32034 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) | |
10 | 2, 9 | mpan 706 | . . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) |
11 | 8, 10 | mpbiri 248 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦Image𝑅(𝑅 “ 𝑦)) |
12 | breq2 4657 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦))) | |
13 | 12 | spcegv 3294 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) → ∃𝑥 𝑦Image𝑅𝑥)) |
14 | 11, 13 | mpd 15 | . . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥) |
15 | 7, 14 | impbii 199 | . . . 4 ⊢ (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅 “ 𝑦) ∈ V) |
16 | 2 | eldm 5321 | . . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥) |
17 | imaeq2 5462 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦)) | |
18 | 17 | eleq1d 2686 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V)) |
19 | 2, 18 | elab 3350 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V) |
20 | 15, 16, 19 | 3bitr4i 292 | . . 3 ⊢ (𝑦 ∈ dom Image𝑅 ↔ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}) |
21 | 20 | eqriv 2619 | . 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
22 | df-fn 5891 | . 2 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})) | |
23 | 1, 21, 22 | mpbir2an 955 | 1 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cab 2608 Vcvv 3200 class class class wbr 4653 dom cdm 5114 “ cima 5117 Fun wfun 5882 Fn wfn 5883 Imagecimage 31947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-symdif 3844 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-eprel 5029 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-image 31971 |
This theorem is referenced by: imageval 32037 |
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