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Mathbox for Scott Fenton |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brimg | Structured version Visualization version Unicode version | ||
| Description: Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| brimg.1 |
    |
| brimg.2 |
 |
| brimg.3 |
 |
| Ref | Expression |
|---|---|
| brimg |
  
 Img     |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-img 31973 |
. . 3
Img Image  
  Cart | |
| 2 | 1 | breqi 4659 |
. 2
Img Image
Cart |
| 3 | opex 4932 |
. . . 4
| |
| 4 | brimg.3 |
. . . 4
| |
| 5 | 3, 4 | brco 5292 |
. . 3
Image
Cart
 Cart ![/\](wedge.gif "/\"){kind=small} Image
|
| 6 | brimg.1 |
. . . . . 6
| |
| 7 | brimg.2 |
. . . . . 6
| |
| 8 | vex 3203 |
. . . . . 6
| |
| 9 | 6, 7, 8 | brcart 32039 |
. . . . 5
Cart |
| 10 | 9 | anbi1i 731 |
. . . 4
Cart Image
Image
|
| 11 | 10 | exbii 1774 |
. . 3
Cart Image
Image
|
| 12 | 6, 7 | xpex 6962 |
. . . 4
|
| 13 | breq1 4656 |
. . . 4
 Image Image | |
| 14 | 12, 13 | ceqsexv 3242 |
. . 3
Image
Image |
| 15 | 5, 11, 14 | 3bitri 286 |
. 2
Image
Cart
Image |
| 16 | 12, 4 | brimage 32033 |
. . 3
Image
|
| 17 | 19.42v 1918 |
. . . . . . . 8
| |
| 18 | anass 681 |
. . . . . . . . . . 11
| |
| 19 | anass 681 |
. . . . . . . . . . . . 13
| |
| 20 | an12 838 |
. . . . . . . . . . . . 13
| |
| 21 | 19, 20 | bitri 264 |
. . . . . . . . . . . 12
|
| 22 | 21 | anbi2i 730 |
. . . . . . . . . . 11
|
| 23 | 18, 22 | bitri 264 |
. . . . . . . . . 10
|
| 24 | 23 | 2exbii 1775 |
. . . . . . . . 9
|
| 25 | excom 2042 |
. . . . . . . . 9
| |
| 26 | opex 4932 |
. . . . . . . . . . 11
| |
| 27 | breq1 4656 |
. . . . . . . . . . . . 13
| |
| 28 | 27 | anbi2d 740 |
. . . . . . . . . . . 12
|
| 29 | 28 | anbi2d 740 |
. . . . . . . . . . 11
|
| 30 | 26, 29 | ceqsexv 3242 |
. . . . . . . . . 10
|
| 31 | 30 | exbii 1774 |
. . . . . . . . 9
|
| 32 | 24, 25, 31 | 3bitri 286 |
. . . . . . . 8
|
| 33 | df-br 4654 |
. . . . . . . . . 10
| |
| 34 | risset 3062 |
. . . . . . . . . . 11
| |
| 35 | vex 3203 |
. . . . . . . . . . . . . . . 16
| |
| 36 | 35 | brres 5402 |
. . . . . . . . . . . . . . 15
|
| 37 | df-br 4654 |
. . . . . . . . . . . . . . 15
| |
| 38 | ancom 466 |
. . . . . . . . . . . . . . 15
| |
| 39 | 36, 37, 38 | 3bitr3ri 291 |
. . . . . . . . . . . . . 14
|
| 40 | 39 | anbi2i 730 |
. . . . . . . . . . . . 13
|
| 41 | elvv 5177 |
. . . . . . . . . . . . . . . 16
| |
| 42 | 41 | anbi1i 731 |
. . . . . . . . . . . . . . 15
|
| 43 | anass 681 |
. . . . . . . . . . . . . . 15
| |
| 44 | ancom 466 |
. . . . . . . . . . . . . . . . . 18
| |
| 45 | breq1 4656 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 46 | opeq1 4402 |
. . . . . . . . . . . . . . . . . . . . . 22
| |
| 47 | 46 | breq1d 4663 |
. . . . . . . . . . . . . . . . . . . . 21
|
| 48 | 45, 47 | anbi12d 747 |
. . . . . . . . . . . . . . . . . . . 20
|
| 49 | vex 3203 |
. . . . . . . . . . . . . . . . . . . . . . 23
| |
| 50 | vex 3203 |
. . . . . . . . . . . . . . . . . . . . . . 23
| |
| 51 | 49, 50 | br1steq 31670 |
. . . . . . . . . . . . . . . . . . . . . 22
|
| 52 | equcom 1945 |
. . . . . . . . . . . . . . . . . . . . . 22
| |
| 53 | 51, 52 | bitri 264 |
. . . . . . . . . . . . . . . . . . . . 21
|
| 54 | opex 4932 |
. . . . . . . . . . . . . . . . . . . . . . . 24
| |
| 55 | vex 3203 |
. . . . . . . . . . . . . . . . . . . . . . . 24
| |
| 56 | 54, 55 | brco 5292 |
. . . . . . . . . . . . . . . . . . . . . . 23
|
| 57 | opex 4932 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
| |
| 58 | 57, 35 | br1steq 31670 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
|
| 59 | 58 | anbi1i 731 |
. . . . . . . . . . . . . . . . . . . . . . . 24
|
| 60 | 59 | exbii 1774 |
. . . . . . . . . . . . . . . . . . . . . . 23
|
| 61 | 56, 60 | bitri 264 |
. . . . . . . . . . . . . . . . . . . . . 22
|
| 62 | breq1 4656 |
. . . . . . . . . . . . . . . . . . . . . . . 24
| |
| 63 | 57, 62 | ceqsexv 3242 |
. . . . . . . . . . . . . . . . . . . . . . 23
|
| 64 | 49, 50 | br2ndeq 31671 |
. . . . . . . . . . . . . . . . . . . . . . 23
|
| 65 | 63, 64 | bitri 264 |
. . . . . . . . . . . . . . . . . . . . . 22
|
| 66 | equcom 1945 |
. . . . . . . . . . . . . . . . . . . . . 22
| |
| 67 | 61, 65, 66 | 3bitri 286 |
. . . . . . . . . . . . . . . . . . . . 21
|
| 68 | 53, 67 | anbi12i 733 |
. . . . . . . . . . . . . . . . . . . 20
|
| 69 | 48, 68 | syl6bb 276 |
. . . . . . . . . . . . . . . . . . 19
|
| 70 | 69 | pm5.32i 669 |
. . . . . . . . . . . . . . . . . 18
|
| 71 | df-3an 1039 |
. . . . . . . . . . . . . . . . . 18
| |
| 72 | 44, 70, 71 | 3bitr4i 292 |
. . . . . . . . . . . . . . . . 17
|
| 73 | 72 | 2exbii 1775 |
. . . . . . . . . . . . . . . 16
|
| 74 | 19.41vv 1915 |
. . . . . . . . . . . . . . . 16
| |
| 75 | opeq1 4402 |
. . . . . . . . . . . . . . . . . 18
| |
| 76 | 75 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . 17
|
| 77 | opeq2 4403 |
. . . . . . . . . . . . . . . . . 18
| |
| 78 | 77 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . 17
|
| 79 | 35, 55, 76, 78 | ceqsex2v 3245 |
. . . . . . . . . . . . . . . 16
|
| 80 | 73, 74, 79 | 3bitr3ri 291 |
. . . . . . . . . . . . . . 15
|
| 81 | 42, 43, 80 | 3bitr4ri 293 |
. . . . . . . . . . . . . 14
|
| 82 | ancom 466 |
. . . . . . . . . . . . . 14
| |
| 83 | 81, 82 | bitri 264 |
. . . . . . . . . . . . 13
|
| 84 | 55 | brres 5402 |
. . . . . . . . . . . . 13
|
| 85 | 40, 83, 84 | 3bitr4i 292 |
. . . . . . . . . . . 12
|
| 86 | 85 | rexbii 3041 |
. . . . . . . . . . 11
|
| 87 | 34, 86 | bitri 264 |
. . . . . . . . . 10
|
| 88 | df-rex 2918 |
. . . . . . . . . 10
| |
| 89 | 33, 87, 88 | 3bitri 286 |
. . . . . . . . 9
|
| 90 | 89 | anbi2i 730 |
. . . . . . . 8
|
| 91 | 17, 32, 90 | 3bitr4ri 293 |
. . . . . . 7
|
| 92 | 91 | exbii 1774 |
. . . . . 6
|
| 93 | 55 | elima2 5472 |
. . . . . 6
|
| 94 | 55 | elima2 5472 |
. . . . . . 7
|
| 95 | elxp 5131 |
. . . . . . . . . 10
| |
| 96 | 95 | anbi1i 731 |
. . . . . . . . 9
|
| 97 | 19.41vv 1915 |
. . . . . . . . 9
| |
| 98 | 96, 97 | bitr4i 267 |
. . . . . . . 8
|
| 99 | 98 | exbii 1774 |
. . . . . . 7
|
| 100 | exrot3 2045 |
. . . . . . . 8
| |
| 101 | exrot3 2045 |
. . . . . . . 8
| |
| 102 | 100, 101 | bitri 264 |
. . . . . . 7
|
| 103 | 94, 99, 102 | 3bitri 286 |
. . . . . 6
|
| 104 | 92, 93, 103 | 3bitr4ri 293 |
. . . . 5
|
| 105 | 104 | eqriv 2619 |
. . . 4
|
| 106 | 105 | eqeq2i 2634 |
. . 3
|
| 107 | 16, 106 | bitri 264 |
. 2
Image
|
| 108 | 2, 15, 107 | 3bitri 286 |
1
Img |
| Colors of variables: wff setvar class |
| Syntax hints:
wb 196
wa 384
w3a 1037
wceq 1483
wex 1704
wcel 1990
wrex 2913
cvv 3200
cop 4183
class class class wbr 4653 cxp 5112 cres 5116 cima 5117 ccom 5118 c1st 7166 c2nd 7167 Imagecimage 31947 Cartccart 31948 Imgcimg 31949 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-symdif 3844 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-eprel 5029 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-pprod 31962 df-image 31971 df-cart 31972 df-img 31973 |
| This theorem is referenced by: brapply 32045 dfrdg4 32058 |
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