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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 32846 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
4 | brxp 5454 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 698 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | vex 3418 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3418 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | brcnv 5604 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
9 | 8 | rexbii 3194 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
10 | 6, 1 | coep 32507 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
11 | 6 | elima 5777 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
12 | 9, 10, 11 | 3bitr4ri 296 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴) |
13 | 1, 2, 3, 5, 12 | brtxpsd3 32878 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 ∃wrex 3089 Vcvv 3415 class class class wbr 4930 E cep 5317 × cxp 5406 ◡ccnv 5407 “ cima 5411 ∘ ccom 5412 Imagecimage 32822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-symdif 4108 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-eprel 5318 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-fo 6196 df-fv 6198 df-1st 7503 df-2nd 7504 df-txp 32836 df-image 32846 |
This theorem is referenced by: brimageg 32909 funimage 32910 fnimage 32911 imageval 32912 brdomain 32915 brrange 32916 brimg 32919 funpartlem 32924 imagesset 32935 |
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