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Mirrors > Home > MPE Home > Th. List > invfun | Structured version Visualization version Unicode version |
Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b |
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invfval.n |
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invfval.c |
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invfval.x |
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invfval.y |
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Ref | Expression |
---|---|
invfun |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b |
. . . 4
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2 | invfval.n |
. . . 4
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3 | invfval.c |
. . . 4
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4 | invfval.x |
. . . 4
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5 | invfval.y |
. . . 4
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6 | eqid 2453 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 3, 4, 5, 6 | invss 15678 |
. . 3
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8 | relxp 4945 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | relss 4925 |
. . 3
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10 | 7, 8, 9 | mpisyl 21 |
. 2
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11 | eqid 2453 |
. . . . . 6
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12 | 3 | adantr 467 |
. . . . . 6
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13 | 5 | adantr 467 |
. . . . . 6
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14 | 4 | adantr 467 |
. . . . . 6
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15 | 1, 2, 3, 4, 5, 11 | isinv 15677 |
. . . . . . . 8
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16 | 15 | simplbda 630 |
. . . . . . 7
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17 | 16 | adantrr 724 |
. . . . . 6
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18 | 1, 2, 3, 4, 5, 11 | isinv 15677 |
. . . . . . . 8
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19 | 18 | simprbda 629 |
. . . . . . 7
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20 | 19 | adantrl 723 |
. . . . . 6
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21 | 1, 11, 12, 13, 14, 17, 20 | sectcan 15672 |
. . . . 5
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22 | 21 | ex 436 |
. . . 4
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23 | 22 | alrimiv 1775 |
. . 3
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24 | 23 | alrimivv 1776 |
. 2
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25 | dffun2 5595 |
. 2
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26 | 10, 24, 25 | sylanbrc 671 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-1st 6798 df-2nd 6799 df-cat 15586 df-cid 15587 df-sect 15664 df-inv 15665 |
This theorem is referenced by: inviso1 15683 invf 15685 invco 15688 idinv 15706 funciso 15791 ffthiso 15846 fuciso 15892 setciso 15998 catciso 16014 rngciso 40088 rngcisoALTV 40100 ringciso 40139 ringcisoALTV 40163 |
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