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Mirrors > Home > MPE Home > Th. List > Mathboxes > brimage | Structured version Visualization version GIF version |
Description: Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brimage.1 | ⊢ 𝐴 ∈ V |
brimage.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brimage | ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brimage.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | brimage.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | df-image 35845 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
4 | brxp 5737 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 711 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | vex 3481 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3481 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | brcnv 5895 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
9 | 8 | rexbii 3091 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
10 | 6, 1 | coep 35731 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
11 | 6 | elima 6084 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
12 | 9, 10, 11 | 3bitr4ri 304 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴) |
13 | 1, 2, 3, 5, 12 | brtxpsd3 35877 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 class class class wbr 5147 E cep 5587 × cxp 5686 ◡ccnv 5687 “ cima 5691 ∘ ccom 5692 Imagecimage 35821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-symdif 4258 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-eprel 5588 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-1st 8012 df-2nd 8013 df-txp 35835 df-image 35845 |
This theorem is referenced by: brimageg 35908 funimage 35909 fnimage 35910 imageval 35911 brdomain 35914 brrange 35915 brimg 35918 funpartlem 35923 imagesset 35934 |
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