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Mirrors > Home > MPE Home > Th. List > ginvsn | Structured version Unicode version |
Description: The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ablsn.1 |
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Ref | Expression |
---|---|
ginvsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsn.1 |
. . . . 5
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2 | fvi 5852 |
. . . . 5
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3 | 1, 2 | ax-mp 5 |
. . . 4
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4 | 3 | opeq2i 4166 |
. . 3
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5 | 4 | sneqi 3991 |
. 2
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6 | f1ovi 5780 |
. . . 4
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7 | f1of 5744 |
. . . 4
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8 | ffn 5662 |
. . . 4
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9 | 6, 7, 8 | mp2b 10 |
. . 3
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10 | fnressn 5998 |
. . 3
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11 | 9, 1, 10 | mp2an 672 |
. 2
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12 | 1 | grposn 23849 |
. . . 4
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13 | opex 4659 |
. . . . . . 7
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14 | 13 | rnsnop 5423 |
. . . . . 6
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15 | 14 | eqcomi 2465 |
. . . . 5
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16 | eqid 2452 |
. . . . 5
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17 | 15, 16 | grpoinvf 23874 |
. . . 4
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18 | f1of 5744 |
. . . 4
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19 | 12, 17, 18 | mp2b 10 |
. . 3
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20 | 1, 1 | fsn 5985 |
. . 3
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21 | 19, 20 | mpbi 208 |
. 2
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22 | 5, 11, 21 | 3eqtr4ri 2492 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-rep 4506 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-reu 2803 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-iun 4276 df-br 4396 df-opab 4454 df-mpt 4455 df-id 4739 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-riota 6156 df-ov 6198 df-grpo 23825 df-gid 23826 df-ginv 23827 |
This theorem is referenced by: (None) |
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