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Mirrors > Home > ILE Home > Th. List > f1opw2 | Unicode version |
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5985 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
f1opw2.1 |
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f1opw2.2 |
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f1opw2.3 |
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Ref | Expression |
---|---|
f1opw2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 |
. 2
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2 | imassrn 4900 |
. . . . 5
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3 | f1opw2.1 |
. . . . . . 7
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4 | f1ofo 5382 |
. . . . . . 7
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5 | 3, 4 | syl 14 |
. . . . . 6
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6 | forn 5356 |
. . . . . 6
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7 | 5, 6 | syl 14 |
. . . . 5
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8 | 2, 7 | sseqtrid 3152 |
. . . 4
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9 | f1opw2.3 |
. . . . 5
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10 | elpwg 3523 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 8, 11 | mpbird 166 |
. . 3
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13 | 12 | adantr 274 |
. 2
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14 | imassrn 4900 |
. . . . 5
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15 | dfdm4 4739 |
. . . . . 6
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16 | f1odm 5379 |
. . . . . . 7
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17 | 3, 16 | syl 14 |
. . . . . 6
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18 | 15, 17 | syl5eqr 2187 |
. . . . 5
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19 | 14, 18 | sseqtrid 3152 |
. . . 4
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20 | f1opw2.2 |
. . . . 5
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21 | elpwg 3523 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 19, 22 | mpbird 166 |
. . 3
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24 | 23 | adantr 274 |
. 2
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25 | elpwi 3524 |
. . . . . . 7
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26 | 25 | adantl 275 |
. . . . . 6
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27 | foimacnv 5393 |
. . . . . 6
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28 | 5, 26, 27 | syl2an 287 |
. . . . 5
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29 | 28 | eqcomd 2146 |
. . . 4
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30 | imaeq2 4885 |
. . . . 5
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31 | 30 | eqeq2d 2152 |
. . . 4
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32 | 29, 31 | syl5ibrcom 156 |
. . 3
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33 | f1of1 5374 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 3, 33 | syl 14 |
. . . . . 6
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35 | elpwi 3524 |
. . . . . . 7
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36 | 35 | adantr 274 |
. . . . . 6
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37 | f1imacnv 5392 |
. . . . . 6
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38 | 34, 36, 37 | syl2an 287 |
. . . . 5
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39 | 38 | eqcomd 2146 |
. . . 4
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40 | imaeq2 4885 |
. . . . 5
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41 | 40 | eqeq2d 2152 |
. . . 4
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42 | 39, 41 | syl5ibrcom 156 |
. . 3
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43 | 32, 42 | impbid 128 |
. 2
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44 | 1, 13, 24, 43 | f1o2d 5983 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 |
This theorem is referenced by: f1opw 5985 |
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