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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 4892 | . . . . . 6 | |
2 | dfdm4 4731 | . . . . . . 7 | |
3 | fof 5345 | . . . . . . . 8 | |
4 | fdm 5278 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | syl5eqr 2186 | . . . . . 6 |
7 | 1, 6 | sseqtrid 3147 | . . . . 5 |
8 | 7 | adantl 275 | . . . 4 |
9 | cnvexg 5076 | . . . . . 6 | |
10 | 9 | adantr 274 | . . . . 5 |
11 | imaexg 4893 | . . . . 5 | |
12 | elpwg 3518 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | 8, 13 | mpbird 166 | . . 3 |
15 | 14 | a1d 22 | . 2 |
16 | imaeq2 4877 | . . . . . . 7 | |
17 | 16 | adantl 275 | . . . . . 6 |
18 | simpllr 523 | . . . . . . 7 | |
19 | simplrl 524 | . . . . . . . 8 | |
20 | 19 | elpwid 3521 | . . . . . . 7 |
21 | foimacnv 5385 | . . . . . . 7 | |
22 | 18, 20, 21 | syl2anc 408 | . . . . . 6 |
23 | simplrr 525 | . . . . . . . 8 | |
24 | 23 | elpwid 3521 | . . . . . . 7 |
25 | foimacnv 5385 | . . . . . . 7 | |
26 | 18, 24, 25 | syl2anc 408 | . . . . . 6 |
27 | 17, 22, 26 | 3eqtr3d 2180 | . . . . 5 |
28 | 27 | ex 114 | . . . 4 |
29 | imaeq2 4877 | . . . 4 | |
30 | 28, 29 | impbid1 141 | . . 3 |
31 | 30 | ex 114 | . 2 |
32 | rnexg 4804 | . . . . 5 | |
33 | forn 5348 | . . . . . 6 | |
34 | 33 | eleq1d 2208 | . . . . 5 |
35 | 32, 34 | syl5ibcom 154 | . . . 4 |
36 | 35 | imp 123 | . . 3 |
37 | pwexg 4104 | . . 3 | |
38 | 36, 37 | syl 14 | . 2 |
39 | dmfex 5312 | . . . 4 | |
40 | 3, 39 | sylan2 284 | . . 3 |
41 | pwexg 4104 | . . 3 | |
42 | 40, 41 | syl 14 | . 2 |
43 | 15, 31, 38, 42 | dom3d 6668 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 wss 3071 cpw 3510 class class class wbr 3929 ccnv 4538 cdm 4539 crn 4540 cima 4542 wf 5119 wfo 5121 cdom 6633 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-fv 5131 df-dom 6636 |
This theorem is referenced by: (None) |
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