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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 5649 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 2037 |
. 2
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2 | imassrn 4622 |
. . . . 5
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3 | f1opw2.1 |
. . . . . . 7
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4 | f1ofo 5076 |
. . . . . . 7
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5 | 3, 4 | syl 14 |
. . . . . 6
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6 | forn 5052 |
. . . . . 6
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7 | 5, 6 | syl 14 |
. . . . 5
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8 | 2, 7 | syl5sseq 2987 |
. . . 4
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9 | f1opw2.3 |
. . . . 5
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10 | elpwg 3359 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 8, 11 | mpbird 156 |
. . 3
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13 | 12 | adantr 261 |
. 2
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14 | imassrn 4622 |
. . . . 5
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15 | dfdm4 4470 |
. . . . . 6
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16 | f1odm 5073 |
. . . . . . 7
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17 | 3, 16 | syl 14 |
. . . . . 6
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18 | 15, 17 | syl5eqr 2083 |
. . . . 5
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19 | 14, 18 | syl5sseq 2987 |
. . . 4
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20 | f1opw2.2 |
. . . . 5
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21 | elpwg 3359 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 19, 22 | mpbird 156 |
. . 3
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24 | 23 | adantr 261 |
. 2
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25 | elpwi 3360 |
. . . . . . 7
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26 | 25 | adantl 262 |
. . . . . 6
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27 | foimacnv 5087 |
. . . . . 6
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28 | 5, 26, 27 | syl2an 273 |
. . . . 5
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29 | 28 | eqcomd 2042 |
. . . 4
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30 | imaeq2 4607 |
. . . . 5
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31 | 30 | eqeq2d 2048 |
. . . 4
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32 | 29, 31 | syl5ibrcom 146 |
. . 3
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33 | f1of1 5068 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 3, 33 | syl 14 |
. . . . . 6
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35 | elpwi 3360 |
. . . . . . 7
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36 | 35 | adantr 261 |
. . . . . 6
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37 | f1imacnv 5086 |
. . . . . 6
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38 | 34, 36, 37 | syl2an 273 |
. . . . 5
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39 | 38 | eqcomd 2042 |
. . . 4
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40 | imaeq2 4607 |
. . . . 5
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41 | 40 | eqeq2d 2048 |
. . . 4
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42 | 39, 41 | syl5ibrcom 146 |
. . 3
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43 | 32, 42 | impbid 120 |
. 2
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44 | 1, 13, 24, 43 | f1o2d 5647 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: f1opw 5649 |
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