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Mirrors > Home > ILE Home > Th. List > cnvco | Unicode version |
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1608 |
. . . 4
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2 | vex 2740 |
. . . . 5
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3 | vex 2740 |
. . . . 5
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4 | 2, 3 | brco 4797 |
. . . 4
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5 | vex 2740 |
. . . . . . 7
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6 | 3, 5 | brcnv 4809 |
. . . . . 6
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7 | 5, 2 | brcnv 4809 |
. . . . . 6
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8 | 6, 7 | anbi12i 460 |
. . . . 5
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9 | 8 | exbii 1605 |
. . . 4
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10 | 1, 4, 9 | 3bitr4i 212 |
. . 3
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11 | 10 | opabbii 4069 |
. 2
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12 | df-cnv 4633 |
. 2
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13 | df-co 4634 |
. 2
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14 | 11, 12, 13 | 3eqtr4i 2208 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-cnv 4633 df-co 4634 |
This theorem is referenced by: rncoss 4896 rncoeq 4899 dmco 5136 cores2 5140 co01 5142 coi2 5144 relcnvtr 5147 dfdm2 5162 f1co 5432 cofunex2g 6108 caseinj 7085 djuinj 7102 cnco 13592 hmeoco 13687 |
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