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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 6644 |
. . . 4
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2 | 1 | elmpocl1 6063 |
. . 3
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3 | 2 | adantl 277 |
. 2
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4 | simpl 109 |
. 2
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5 | elpmi 6660 |
. . . . . 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 3355 |
. . . 4
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9 | fssres 5386 |
. . . 4
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10 | 7, 8, 9 | sylancl 413 |
. . 3
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11 | ffun 5363 |
. . . . 5
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12 | resres 4914 |
. . . . . 6
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13 | funrel 5228 |
. . . . . . 7
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14 | resdm 4941 |
. . . . . . 7
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15 | reseq1 4896 |
. . . . . . 7
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16 | 13, 14, 15 | 3syl 17 |
. . . . . 6
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17 | 12, 16 | eqtr3id 2224 |
. . . . 5
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18 | 7, 11, 17 | 3syl 17 |
. . . 4
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19 | 18 | feq1d 5347 |
. . 3
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20 | 10, 19 | mpbid 147 |
. 2
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21 | inss2 3356 |
. . 3
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22 | elpm2r 6659 |
. . 3
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23 | 21, 22 | mpanr2 438 |
. 2
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24 | 3, 4, 20, 23 | syl21anc 1237 |
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 614 ax-in2 615 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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pm 6644 |
This theorem is referenced by: lmres 13381 |
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