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Mirrors > Home > MPE Home > Th. List > Mathboxes > brimage | Structured version Visualization version GIF version |
Description: Binary relationship 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 31140 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
4 | brxp 5071 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 957 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | brcnv 5227 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
9 | 8 | rexbii 3023 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
10 | 6, 1 | coep 30894 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
11 | 6 | elima 5390 | . . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
12 | 9, 10, 11 | 3bitr4ri 292 | . 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴) |
13 | 1, 2, 3, 5, 12 | brtxpsd3 31173 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 E cep 4947 × cxp 5036 ◡ccnv 5037 “ cima 5041 ∘ ccom 5042 Imagecimage 31116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-image 31140 |
This theorem is referenced by: brimageg 31204 funimage 31205 fnimage 31206 imageval 31207 brdomain 31210 brrange 31211 brimg 31214 funpartlem 31219 imagesset 31230 |
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