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Mirrors > Home > ILE Home > Th. List > isoini2 | Unicode version |
Description: Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Ref | Expression |
---|---|
isoini2.1 |
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isoini2.2 |
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Ref | Expression |
---|---|
isoini2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isof1o 5586 |
. . . . . 6
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2 | f1of1 5252 |
. . . . . 6
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3 | 1, 2 | syl 14 |
. . . . 5
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4 | 3 | adantr 270 |
. . . 4
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5 | isoini2.1 |
. . . . 5
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6 | inss1 3220 |
. . . . 5
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7 | 5, 6 | eqsstri 3056 |
. . . 4
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8 | f1ores 5268 |
. . . 4
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9 | 4, 7, 8 | sylancl 404 |
. . 3
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10 | isoini 5597 |
. . . . 5
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11 | 5 | imaeq2i 4772 |
. . . . 5
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12 | isoini2.2 |
. . . . 5
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13 | 10, 11, 12 | 3eqtr4g 2145 |
. . . 4
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14 | f1oeq3 5246 |
. . . 4
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15 | 13, 14 | syl 14 |
. . 3
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16 | 9, 15 | mpbid 145 |
. 2
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17 | df-isom 5024 |
. . . . . . 7
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18 | 17 | simprbi 269 |
. . . . . 6
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19 | 18 | adantr 270 |
. . . . 5
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20 | ssralv 3085 |
. . . . . 6
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21 | 20 | ralimdv 2442 |
. . . . 5
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22 | 7, 19, 21 | mpsyl 64 |
. . . 4
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23 | ssralv 3085 |
. . . 4
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24 | 7, 22, 23 | mpsyl 64 |
. . 3
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25 | fvres 5329 |
. . . . . . 7
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26 | fvres 5329 |
. . . . . . 7
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27 | 25, 26 | breqan12d 3860 |
. . . . . 6
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28 | 27 | bibi2d 230 |
. . . . 5
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29 | 28 | ralbidva 2376 |
. . . 4
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30 | 29 | ralbiia 2392 |
. . 3
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31 | 24, 30 | sylibr 132 |
. 2
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32 | df-isom 5024 |
. 2
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33 | 16, 31, 32 | sylanbrc 408 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-isom 5024 |
This theorem is referenced by: (None) |
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