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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 4481 |
. . . . . 6
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2 | cnvcnv 4898 |
. . . . . . . 8
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3 | inss2 3224 |
. . . . . . . 8
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4 | 2, 3 | eqsstri 3059 |
. . . . . . 7
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5 | 4 | sseli 3024 |
. . . . . 6
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6 | 1, 5 | mto 624 |
. . . . 5
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7 | disjsn 3510 |
. . . . 5
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8 | 6, 7 | mpbir 145 |
. . . 4
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9 | cnvcnvss 4900 |
. . . . 5
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10 | reldisj 3340 |
. . . . 5
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11 | 9, 10 | ax-mp 7 |
. . . 4
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12 | 8, 11 | mpbi 144 |
. . 3
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13 | 12 | a1i 9 |
. 2
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14 | structn0fun 11570 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | funrel 5047 |
. . . . 5
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16 | 14, 15 | syl 14 |
. . . 4
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17 | dfrel2 4896 |
. . . 4
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18 | 16, 17 | sylib 121 |
. . 3
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19 | difss 3129 |
. . . 4
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20 | cnvss 4624 |
. . . 4
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21 | cnvss 4624 |
. . . 4
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22 | 19, 20, 21 | mp2b 8 |
. . 3
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23 | 18, 22 | syl6eqssr 3080 |
. 2
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24 | 13, 23 | eqssd 3045 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-struct 11559 |
This theorem is referenced by: structfung 11574 |
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