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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdm5 | Structured version Visualization version Unicode version |
Description: Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
dfdm5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2042 | . . . 4 | |
2 | opex 4932 | . . . . . . . 8 | |
3 | breq1 4656 | . . . . . . . . . 10 | |
4 | eleq1 2689 | . . . . . . . . . 10 | |
5 | 3, 4 | anbi12d 747 | . . . . . . . . 9 |
6 | vex 3203 | . . . . . . . . . . . 12 | |
7 | vex 3203 | . . . . . . . . . . . 12 | |
8 | 6, 7 | br1steq 31670 | . . . . . . . . . . 11 |
9 | equcom 1945 | . . . . . . . . . . 11 | |
10 | 8, 9 | bitri 264 | . . . . . . . . . 10 |
11 | 10 | anbi1i 731 | . . . . . . . . 9 |
12 | 5, 11 | syl6bb 276 | . . . . . . . 8 |
13 | 2, 12 | ceqsexv 3242 | . . . . . . 7 |
14 | 13 | exbii 1774 | . . . . . 6 |
15 | excom 2042 | . . . . . 6 | |
16 | vex 3203 | . . . . . . 7 | |
17 | opeq1 4402 | . . . . . . . 8 | |
18 | 17 | eleq1d 2686 | . . . . . . 7 |
19 | 16, 18 | ceqsexv 3242 | . . . . . 6 |
20 | 14, 15, 19 | 3bitr3ri 291 | . . . . 5 |
21 | 20 | exbii 1774 | . . . 4 |
22 | ancom 466 | . . . . . 6 | |
23 | anass 681 | . . . . . . 7 | |
24 | 16 | brres 5402 | . . . . . . . . 9 |
25 | ancom 466 | . . . . . . . . . 10 | |
26 | elvv 5177 | . . . . . . . . . . . 12 | |
27 | excom 2042 | . . . . . . . . . . . 12 | |
28 | 26, 27 | bitri 264 | . . . . . . . . . . 11 |
29 | 28 | anbi1i 731 | . . . . . . . . . 10 |
30 | 25, 29 | bitri 264 | . . . . . . . . 9 |
31 | 24, 30 | bitri 264 | . . . . . . . 8 |
32 | 31 | anbi1i 731 | . . . . . . 7 |
33 | 19.41vv 1915 | . . . . . . 7 | |
34 | 23, 32, 33 | 3bitr4i 292 | . . . . . 6 |
35 | 22, 34 | bitri 264 | . . . . 5 |
36 | 35 | exbii 1774 | . . . 4 |
37 | 1, 21, 36 | 3bitr4i 292 | . . 3 |
38 | 16 | eldm2 5322 | . . 3 |
39 | 16 | elima2 5472 | . . 3 |
40 | 37, 38, 39 | 3bitr4i 292 | . 2 |
41 | 40 | eqriv 2619 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 cop 4183 class class class wbr 4653 cxp 5112 cdm 5114 cres 5116 cima 5117 c1st 7166 |
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-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-nul 3916 df-if 4087 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-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 |
This theorem is referenced by: brdomain 32040 |
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