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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 35865 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
| 4 | brxp 5734 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 5 | 1, 2, 4 | mpbir2an 711 | . 2 ⊢ 𝐴(V × V)𝐵 |
| 6 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 7 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 8 | 6, 7 | brcnv 5893 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
| 9 | 8 | rexbii 3094 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
| 10 | 6, 1 | coep 35752 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
| 11 | 6 | elima 6083 | . . 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 35897 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 E cep 5583 × cxp 5683 ◡ccnv 5684 “ cima 5688 ∘ ccom 5689 Imagecimage 35841 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-image 35865 |
| This theorem is referenced by: brimageg 35928 funimage 35929 fnimage 35930 imageval 35931 brdomain 35934 brrange 35935 brimg 35938 funpartlem 35943 imagesset 35954 |
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