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Mirrors > Home > MPE Home > Th. List > f1opw2 | Structured version Visualization version Unicode version |
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6520 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 2450 |
. 2
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2 | imassrn 5178 |
. . . . 5
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3 | f1opw2.1 |
. . . . . . 7
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4 | f1ofo 5819 |
. . . . . . 7
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5 | 3, 4 | syl 17 |
. . . . . 6
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6 | forn 5794 |
. . . . . 6
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7 | 5, 6 | syl 17 |
. . . . 5
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8 | 2, 7 | syl5sseq 3479 |
. . . 4
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9 | f1opw2.3 |
. . . . 5
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10 | elpwg 3958 |
. . . . 5
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11 | 9, 10 | syl 17 |
. . . 4
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12 | 8, 11 | mpbird 236 |
. . 3
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13 | 12 | adantr 467 |
. 2
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14 | imassrn 5178 |
. . . . 5
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15 | dfdm4 5026 |
. . . . . 6
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16 | f1odm 5816 |
. . . . . . 7
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17 | 3, 16 | syl 17 |
. . . . . 6
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18 | 15, 17 | syl5eqr 2498 |
. . . . 5
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19 | 14, 18 | syl5sseq 3479 |
. . . 4
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20 | f1opw2.2 |
. . . . 5
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21 | elpwg 3958 |
. . . . 5
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22 | 20, 21 | syl 17 |
. . . 4
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23 | 19, 22 | mpbird 236 |
. . 3
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24 | 23 | adantr 467 |
. 2
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25 | elpwi 3959 |
. . . . . . 7
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26 | 25 | adantl 468 |
. . . . . 6
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27 | foimacnv 5829 |
. . . . . 6
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28 | 5, 26, 27 | syl2an 480 |
. . . . 5
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29 | 28 | eqcomd 2456 |
. . . 4
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30 | imaeq2 5163 |
. . . . 5
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31 | 30 | eqeq2d 2460 |
. . . 4
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32 | 29, 31 | syl5ibrcom 226 |
. . 3
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33 | f1of1 5811 |
. . . . . . 7
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34 | 3, 33 | syl 17 |
. . . . . 6
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35 | elpwi 3959 |
. . . . . . 7
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36 | 35 | adantr 467 |
. . . . . 6
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37 | f1imacnv 5828 |
. . . . . 6
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38 | 34, 36, 37 | syl2an 480 |
. . . . 5
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39 | 38 | eqcomd 2456 |
. . . 4
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40 | imaeq2 5163 |
. . . . 5
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41 | 40 | eqeq2d 2460 |
. . . 4
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42 | 39, 41 | syl5ibrcom 226 |
. . 3
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43 | 32, 42 | impbid 194 |
. 2
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44 | 1, 13, 24, 43 | f1o2d 6518 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pr 4638 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 |
This theorem is referenced by: f1opw 6520 |
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