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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 35307 | . 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V))) | |
4 | brxp 5725 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 708 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | vex 3477 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3477 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | brcnv 5882 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
9 | 8 | rexbii 3093 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥) |
10 | 6, 1 | coep 35193 | . . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦) |
11 | 6 | elima 6064 | . . 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 35339 | 1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 class class class wbr 5148 E cep 5579 × cxp 5674 ◡ccnv 5675 “ cima 5679 ∘ ccom 5680 Imagecimage 35283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-txp 35297 df-image 35307 |
This theorem is referenced by: brimageg 35370 funimage 35371 fnimage 35372 imageval 35373 brdomain 35376 brrange 35377 brimg 35380 funpartlem 35385 imagesset 35396 |
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