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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 34559 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
4 | brxp 5701 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 709 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | vex 3463 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3463 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | brcnv 5858 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
9 | 8 | rexbii 3093 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
10 | 6, 1 | coep 34445 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
11 | 6 | elima 6038 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
12 | 9, 10, 11 | 3bitr4ri 303 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴) |
13 | 1, 2, 3, 5, 12 | brtxpsd3 34591 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 Vcvv 3459 class class class wbr 5125 E cep 5556 × cxp 5651 ◡ccnv 5652 “ cima 5656 ∘ ccom 5657 Imagecimage 34535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-symdif 4222 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-eprel 5557 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-fo 6522 df-fv 6524 df-1st 7941 df-2nd 7942 df-txp 34549 df-image 34559 |
This theorem is referenced by: brimageg 34622 funimage 34623 fnimage 34624 imageval 34625 brdomain 34628 brrange 34629 brimg 34632 funpartlem 34637 imagesset 34648 |
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