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Mirrors > Home > ILE Home > Th. List > structcnvcnv | Unicode version |
Description: Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
structcnvcnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4668 |
. . . . . 6
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2 | cnvcnv 5095 |
. . . . . . . 8
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3 | inss2 3370 |
. . . . . . . 8
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4 | 2, 3 | eqsstri 3201 |
. . . . . . 7
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5 | 4 | sseli 3165 |
. . . . . 6
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6 | 1, 5 | mto 663 |
. . . . 5
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7 | disjsn 3668 |
. . . . 5
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8 | 6, 7 | mpbir 146 |
. . . 4
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9 | cnvcnvss 5097 |
. . . . 5
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10 | reldisj 3488 |
. . . . 5
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11 | 9, 10 | ax-mp 5 |
. . . 4
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12 | 8, 11 | mpbi 145 |
. . 3
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13 | 12 | a1i 9 |
. 2
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14 | structn0fun 12492 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | funrel 5247 |
. . . . 5
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16 | 14, 15 | syl 14 |
. . . 4
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17 | dfrel2 5093 |
. . . 4
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18 | 16, 17 | sylib 122 |
. . 3
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19 | difss 3275 |
. . . 4
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20 | cnvss 4814 |
. . . 4
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21 | cnvss 4814 |
. . . 4
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22 | 19, 20, 21 | mp2b 8 |
. . 3
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23 | 18, 22 | eqsstrrdi 3222 |
. 2
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24 | 13, 23 | eqssd 3186 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-opab 4079 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-struct 12481 |
This theorem is referenced by: structfung 12496 |
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