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Mirrors > Home > ILE Home > Th. List > pmresg | Unicode version |
Description: Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
pmresg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pm 6677 |
. . . 4
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2 | 1 | elmpocl1 6092 |
. . 3
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3 | 2 | adantl 277 |
. 2
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4 | simpl 109 |
. 2
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5 | elpmi 6693 |
. . . . . 6
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6 | 5 | simpld 112 |
. . . . 5
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7 | 6 | adantl 277 |
. . . 4
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8 | inss1 3370 |
. . . 4
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9 | fssres 5410 |
. . . 4
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10 | 7, 8, 9 | sylancl 413 |
. . 3
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11 | ffun 5387 |
. . . . 5
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12 | resres 4937 |
. . . . . 6
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13 | funrel 5252 |
. . . . . . 7
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14 | resdm 4964 |
. . . . . . 7
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15 | reseq1 4919 |
. . . . . . 7
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16 | 13, 14, 15 | 3syl 17 |
. . . . . 6
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17 | 12, 16 | eqtr3id 2236 |
. . . . 5
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18 | 7, 11, 17 | 3syl 17 |
. . . 4
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19 | 18 | feq1d 5371 |
. . 3
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20 | 10, 19 | mpbid 147 |
. 2
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21 | inss2 3371 |
. . 3
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22 | elpm2r 6692 |
. . 3
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23 | 21, 22 | mpanr2 438 |
. 2
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24 | 3, 4, 20, 23 | syl21anc 1248 |
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-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 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-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 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-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pm 6677 |
This theorem is referenced by: lmres 14208 |
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