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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 31205 | . 2 ⊢ Fun Image𝑅 | |
2 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brimage 31203 | . . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦)) |
5 | eqvisset 3184 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V) | |
6 | 4, 5 | sylbi 206 | . . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
7 | 6 | exlimiv 1845 | . . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
8 | eqid 2610 | . . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦) | |
9 | brimageg 31204 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) | |
10 | 2, 9 | mpan 702 | . . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) |
11 | 8, 10 | mpbiri 247 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦Image𝑅(𝑅 “ 𝑦)) |
12 | breq2 4587 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦))) | |
13 | 12 | spcegv 3267 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) → ∃𝑥 𝑦Image𝑅𝑥)) |
14 | 11, 13 | mpd 15 | . . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥) |
15 | 7, 14 | impbii 198 | . . . 4 ⊢ (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅 “ 𝑦) ∈ V) |
16 | 2 | eldm 5243 | . . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥) |
17 | imaeq2 5381 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦)) | |
18 | 17 | eleq1d 2672 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V)) |
19 | 2, 18 | elab 3319 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V) |
20 | 15, 16, 19 | 3bitr4i 291 | . . 3 ⊢ (𝑦 ∈ dom Image𝑅 ↔ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}) |
21 | 20 | eqriv 2607 | . 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
22 | df-fn 5807 | . 2 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})) | |
23 | 1, 21, 22 | mpbir2an 957 | 1 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 Vcvv 3173 class class class wbr 4583 dom cdm 5038 “ cima 5041 Fun wfun 5798 Fn wfn 5799 Imagecimage 31116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-image 31140 |
This theorem is referenced by: imageval 31207 |
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