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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 32624 | . 2 ⊢ Fun Image𝑅 | |
2 | vex 3401 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3401 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brimage 32622 | . . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦)) |
5 | eqvisset 3413 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V) | |
6 | 4, 5 | sylbi 209 | . . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
7 | 6 | exlimiv 1973 | . . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V) |
8 | eqid 2778 | . . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦) | |
9 | brimageg 32623 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) | |
10 | 2, 9 | mpan 680 | . . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦))) |
11 | 8, 10 | mpbiri 250 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦Image𝑅(𝑅 “ 𝑦)) |
12 | breq2 4890 | . . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦))) | |
13 | 12 | spcegv 3496 | . . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) → ∃𝑥 𝑦Image𝑅𝑥)) |
14 | 11, 13 | mpd 15 | . . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥) |
15 | 7, 14 | impbii 201 | . . . 4 ⊢ (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅 “ 𝑦) ∈ V) |
16 | 2 | eldm 5566 | . . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥) |
17 | imaeq2 5716 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦)) | |
18 | 17 | eleq1d 2844 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V)) |
19 | 2, 18 | elab 3558 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V) |
20 | 15, 16, 19 | 3bitr4i 295 | . . 3 ⊢ (𝑦 ∈ dom Image𝑅 ↔ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}) |
21 | 20 | eqriv 2775 | . 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
22 | df-fn 6138 | . 2 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})) | |
23 | 1, 21, 22 | mpbir2an 701 | 1 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∃wex 1823 ∈ wcel 2107 {cab 2763 Vcvv 3398 class class class wbr 4886 dom cdm 5355 “ cima 5358 Fun wfun 6129 Fn wfn 6130 Imagecimage 32536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-symdif 4067 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-eprel 5266 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fo 6141 df-fv 6143 df-1st 7445 df-2nd 7446 df-txp 32550 df-image 32560 |
This theorem is referenced by: imageval 32626 |
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