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Mirrors > Home > ILE Home > Th. List > caserel | Unicode version |
Description: The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
caserel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-case 7114 |
. 2
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2 | cocnvss 5172 |
. . . 4
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3 | inlresf1 7091 |
. . . . . 6
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4 | f1rn 5441 |
. . . . . 6
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5 | 3, 4 | ax-mp 5 |
. . . . 5
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6 | resss 4949 |
. . . . . . 7
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7 | rnss 4875 |
. . . . . . 7
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8 | 6, 7 | ax-mp 5 |
. . . . . 6
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9 | ssun1 3313 |
. . . . . 6
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10 | 8, 9 | sstri 3179 |
. . . . 5
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11 | xpss12 4751 |
. . . . 5
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12 | 5, 10, 11 | mp2an 426 |
. . . 4
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13 | 2, 12 | sstri 3179 |
. . 3
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14 | cocnvss 5172 |
. . . 4
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15 | inrresf1 7092 |
. . . . . 6
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16 | f1rn 5441 |
. . . . . 6
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17 | 15, 16 | ax-mp 5 |
. . . . 5
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18 | resss 4949 |
. . . . . . 7
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19 | rnss 4875 |
. . . . . . 7
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20 | 18, 19 | ax-mp 5 |
. . . . . 6
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21 | ssun2 3314 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | sstri 3179 |
. . . . 5
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23 | xpss12 4751 |
. . . . 5
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24 | 17, 22, 23 | mp2an 426 |
. . . 4
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25 | 14, 24 | sstri 3179 |
. . 3
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26 | 13, 25 | unssi 3325 |
. 2
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27 | 1, 26 | eqsstri 3202 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1st 6166 df-2nd 6167 df-1o 6442 df-dju 7068 df-inl 7077 df-inr 7078 df-case 7114 |
This theorem is referenced by: casef 7118 |
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