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Mirrors > Home > MPE Home > Th. List > isoini2 | Structured version Visualization version 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 6234 |
. . . . . 6
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2 | f1of1 5827 |
. . . . . 6
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3 | 1, 2 | syl 17 |
. . . . 5
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4 | 3 | adantr 472 |
. . . 4
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5 | isoini2.1 |
. . . . 5
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6 | inss1 3643 |
. . . . 5
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7 | 5, 6 | eqsstri 3448 |
. . . 4
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8 | f1ores 5842 |
. . . 4
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9 | 4, 7, 8 | sylancl 675 |
. . 3
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10 | isoini 6247 |
. . . . 5
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11 | 5 | imaeq2i 5172 |
. . . . 5
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12 | isoini2.2 |
. . . . 5
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13 | 10, 11, 12 | 3eqtr4g 2530 |
. . . 4
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14 | f1oeq3 5820 |
. . . 4
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15 | 13, 14 | syl 17 |
. . 3
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16 | 9, 15 | mpbid 215 |
. 2
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17 | df-isom 5598 |
. . . . . . 7
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18 | 17 | simprbi 471 |
. . . . . 6
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19 | 18 | adantr 472 |
. . . . 5
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20 | ssralv 3479 |
. . . . . 6
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21 | 20 | ralimdv 2806 |
. . . . 5
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22 | 7, 19, 21 | mpsyl 64 |
. . . 4
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23 | ssralv 3479 |
. . . 4
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24 | 7, 22, 23 | mpsyl 64 |
. . 3
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25 | fvres 5893 |
. . . . . . 7
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26 | fvres 5893 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | breqan12d 4411 |
. . . . . 6
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28 | 27 | bibi2d 325 |
. . . . 5
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29 | 28 | ralbidva 2828 |
. . . 4
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30 | 29 | ralbiia 2822 |
. . 3
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31 | 24, 30 | sylibr 217 |
. 2
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32 | df-isom 5598 |
. 2
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33 | 16, 31, 32 | sylanbrc 677 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-isom 5598 |
This theorem is referenced by: fz1isolem 12665 |
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