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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 4951 | . . . . . 6 | |
2 | dfdm4 4790 | . . . . . . 7 | |
3 | fof 5404 | . . . . . . . 8 | |
4 | fdm 5337 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | eqtr3id 2211 | . . . . . 6 |
7 | 1, 6 | sseqtrid 3187 | . . . . 5 |
8 | 7 | adantl 275 | . . . 4 |
9 | cnvexg 5135 | . . . . . 6 | |
10 | 9 | adantr 274 | . . . . 5 |
11 | imaexg 4952 | . . . . 5 | |
12 | elpwg 3561 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | 8, 13 | mpbird 166 | . . 3 |
15 | 14 | a1d 22 | . 2 |
16 | imaeq2 4936 | . . . . . . 7 | |
17 | 16 | adantl 275 | . . . . . 6 |
18 | simpllr 524 | . . . . . . 7 | |
19 | simplrl 525 | . . . . . . . 8 | |
20 | 19 | elpwid 3564 | . . . . . . 7 |
21 | foimacnv 5444 | . . . . . . 7 | |
22 | 18, 20, 21 | syl2anc 409 | . . . . . 6 |
23 | simplrr 526 | . . . . . . . 8 | |
24 | 23 | elpwid 3564 | . . . . . . 7 |
25 | foimacnv 5444 | . . . . . . 7 | |
26 | 18, 24, 25 | syl2anc 409 | . . . . . 6 |
27 | 17, 22, 26 | 3eqtr3d 2205 | . . . . 5 |
28 | 27 | ex 114 | . . . 4 |
29 | imaeq2 4936 | . . . 4 | |
30 | 28, 29 | impbid1 141 | . . 3 |
31 | 30 | ex 114 | . 2 |
32 | rnexg 4863 | . . . . 5 | |
33 | forn 5407 | . . . . . 6 | |
34 | 33 | eleq1d 2233 | . . . . 5 |
35 | 32, 34 | syl5ibcom 154 | . . . 4 |
36 | 35 | imp 123 | . . 3 |
37 | pwexg 4153 | . . 3 | |
38 | 36, 37 | syl 14 | . 2 |
39 | dmfex 5371 | . . . 4 | |
40 | 3, 39 | sylan2 284 | . . 3 |
41 | pwexg 4153 | . . 3 | |
42 | 40, 41 | syl 14 | . 2 |
43 | 15, 31, 38, 42 | dom3d 6731 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 wss 3111 cpw 3553 class class class wbr 3976 ccnv 4597 cdm 4598 crn 4599 cima 4601 wf 5178 wfo 5180 cdom 6696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-fv 5190 df-dom 6699 |
This theorem is referenced by: (None) |
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