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Mirrors > Home > ILE Home > Th. List > fopwdom | Unicode version |
Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
fopwdom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 4679 | . . . . . 6 | |
2 | dfdm4 4527 | . . . . . . 7 | |
3 | fof 5106 | . . . . . . . 8 | |
4 | fdm 5050 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | syl5eqr 2086 | . . . . . 6 |
7 | 1, 6 | syl5sseq 2993 | . . . . 5 |
8 | 7 | adantl 262 | . . . 4 |
9 | cnvexg 4855 | . . . . . 6 | |
10 | 9 | adantr 261 | . . . . 5 |
11 | imaexg 4680 | . . . . 5 | |
12 | elpwg 3367 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | 8, 13 | mpbird 156 | . . 3 |
15 | 14 | a1d 22 | . 2 |
16 | imaeq2 4664 | . . . . . . 7 | |
17 | 16 | adantl 262 | . . . . . 6 |
18 | simpllr 486 | . . . . . . 7 | |
19 | simplrl 487 | . . . . . . . 8 | |
20 | 19 | elpwid 3369 | . . . . . . 7 |
21 | foimacnv 5144 | . . . . . . 7 | |
22 | 18, 20, 21 | syl2anc 391 | . . . . . 6 |
23 | simplrr 488 | . . . . . . . 8 | |
24 | 23 | elpwid 3369 | . . . . . . 7 |
25 | foimacnv 5144 | . . . . . . 7 | |
26 | 18, 24, 25 | syl2anc 391 | . . . . . 6 |
27 | 17, 22, 26 | 3eqtr3d 2080 | . . . . 5 |
28 | 27 | ex 108 | . . . 4 |
29 | imaeq2 4664 | . . . 4 | |
30 | 28, 29 | impbid1 130 | . . 3 |
31 | 30 | ex 108 | . 2 |
32 | rnexg 4597 | . . . . 5 | |
33 | forn 5109 | . . . . . 6 | |
34 | 33 | eleq1d 2106 | . . . . 5 |
35 | 32, 34 | syl5ibcom 144 | . . . 4 |
36 | 35 | imp 115 | . . 3 |
37 | pwexg 3933 | . . 3 | |
38 | 36, 37 | syl 14 | . 2 |
39 | dmfex 5079 | . . . 4 | |
40 | 3, 39 | sylan2 270 | . . 3 |
41 | pwexg 3933 | . . 3 | |
42 | 40, 41 | syl 14 | . 2 |
43 | 15, 31, 38, 42 | dom3d 6254 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cvv 2557 wss 2917 cpw 3359 class class class wbr 3764 ccnv 4344 cdm 4345 crn 4346 cima 4348 wf 4898 wfo 4900 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-fv 4910 df-dom 6223 |
This theorem is referenced by: (None) |
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