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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 4699 | . . . . . 6 | |
2 | dfdm4 4545 | . . . . . . 7 | |
3 | fof 5126 | . . . . . . . 8 | |
4 | fdm 5070 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | syl5eqr 2127 | . . . . . 6 |
7 | 1, 6 | syl5sseq 3047 | . . . . 5 |
8 | 7 | adantl 271 | . . . 4 |
9 | cnvexg 4875 | . . . . . 6 | |
10 | 9 | adantr 270 | . . . . 5 |
11 | imaexg 4700 | . . . . 5 | |
12 | elpwg 3390 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | 8, 13 | mpbird 165 | . . 3 |
15 | 14 | a1d 22 | . 2 |
16 | imaeq2 4684 | . . . . . . 7 | |
17 | 16 | adantl 271 | . . . . . 6 |
18 | simpllr 500 | . . . . . . 7 | |
19 | simplrl 501 | . . . . . . . 8 | |
20 | 19 | elpwid 3392 | . . . . . . 7 |
21 | foimacnv 5164 | . . . . . . 7 | |
22 | 18, 20, 21 | syl2anc 403 | . . . . . 6 |
23 | simplrr 502 | . . . . . . . 8 | |
24 | 23 | elpwid 3392 | . . . . . . 7 |
25 | foimacnv 5164 | . . . . . . 7 | |
26 | 18, 24, 25 | syl2anc 403 | . . . . . 6 |
27 | 17, 22, 26 | 3eqtr3d 2121 | . . . . 5 |
28 | 27 | ex 113 | . . . 4 |
29 | imaeq2 4684 | . . . 4 | |
30 | 28, 29 | impbid1 140 | . . 3 |
31 | 30 | ex 113 | . 2 |
32 | rnexg 4615 | . . . . 5 | |
33 | forn 5129 | . . . . . 6 | |
34 | 33 | eleq1d 2147 | . . . . 5 |
35 | 32, 34 | syl5ibcom 153 | . . . 4 |
36 | 35 | imp 122 | . . 3 |
37 | pwexg 3954 | . . 3 | |
38 | 36, 37 | syl 14 | . 2 |
39 | dmfex 5099 | . . . 4 | |
40 | 3, 39 | sylan2 280 | . . 3 |
41 | pwexg 3954 | . . 3 | |
42 | 40, 41 | syl 14 | . 2 |
43 | 15, 31, 38, 42 | dom3d 6277 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cvv 2601 wss 2973 cpw 3382 class class class wbr 3785 ccnv 4362 cdm 4363 crn 4364 cima 4366 wf 4918 wfo 4920 cdom 6243 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-fv 4930 df-dom 6246 |
This theorem is referenced by: (None) |
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