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Mirrors > Home > MPE Home > Th. List > invsym2 | Structured version Unicode version |
Description: The inverse relation is symmetric. (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 |
---|---|
invsym2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b |
. . . . 5
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2 | invfval.n |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | invfval.c |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | invfval.y |
. . . . 5
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5 | invfval.x |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eqid 2451 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 3, 4, 5, 6 | invss 14801 |
. . . 4
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8 | relxp 5045 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | relss 5025 |
. . . 4
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10 | 7, 8, 9 | mpisyl 18 |
. . 3
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11 | relcnv 5304 |
. . 3
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12 | 10, 11 | jctil 537 |
. 2
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13 | 1, 2, 3, 5, 4 | invsym 14802 |
. . . 4
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14 | vex 3071 |
. . . . . 6
![]() ![]() ![]() ![]() | |
15 | vex 3071 |
. . . . . 6
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16 | 14, 15 | brcnv 5120 |
. . . . 5
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17 | df-br 4391 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 16, 17 | bitr3i 251 |
. . . 4
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19 | df-br 4391 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 13, 18, 19 | 3bitr3g 287 |
. . 3
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21 | 20 | eqrelrdv2 5037 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 12, 21 | mpancom 669 |
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 1952 ax-ext 2430 ax-rep 4501 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-1st 6677 df-2nd 6678 df-sect 14788 df-inv 14789 |
This theorem is referenced by: invf 14808 invf1o 14809 invinv 14810 |
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