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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pm 6617 | . . . 4 | |
2 | 1 | elmpocl1 6037 | . . 3 |
3 | 2 | adantl 275 | . 2 |
4 | simpl 108 | . 2 | |
5 | elpmi 6633 | . . . . . 6 | |
6 | 5 | simpld 111 | . . . . 5 |
7 | 6 | adantl 275 | . . . 4 |
8 | inss1 3342 | . . . 4 | |
9 | fssres 5363 | . . . 4 | |
10 | 7, 8, 9 | sylancl 410 | . . 3 |
11 | ffun 5340 | . . . . 5 | |
12 | resres 4896 | . . . . . 6 | |
13 | funrel 5205 | . . . . . . 7 | |
14 | resdm 4923 | . . . . . . 7 | |
15 | reseq1 4878 | . . . . . . 7 | |
16 | 13, 14, 15 | 3syl 17 | . . . . . 6 |
17 | 12, 16 | eqtr3id 2213 | . . . . 5 |
18 | 7, 11, 17 | 3syl 17 | . . . 4 |
19 | 18 | feq1d 5324 | . . 3 |
20 | 10, 19 | mpbid 146 | . 2 |
21 | inss2 3343 | . . 3 | |
22 | elpm2r 6632 | . . 3 | |
23 | 21, 22 | mpanr2 435 | . 2 |
24 | 3, 4, 20, 23 | syl21anc 1227 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1343 wcel 2136 crab 2448 cvv 2726 cin 3115 wss 3116 cpw 3559 cxp 4602 cdm 4604 cres 4606 wrel 4609 wfun 5182 wf 5184 (class class class)co 5842 cpm 6615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pm 6617 |
This theorem is referenced by: lmres 12888 |
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