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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 6608 | . . . 4 | |
2 | 1 | elmpocl1 6031 | . . 3 |
3 | 2 | adantl 275 | . 2 |
4 | simpl 108 | . 2 | |
5 | elpmi 6624 | . . . . . 6 | |
6 | 5 | simpld 111 | . . . . 5 |
7 | 6 | adantl 275 | . . . 4 |
8 | inss1 3337 | . . . 4 | |
9 | fssres 5357 | . . . 4 | |
10 | 7, 8, 9 | sylancl 410 | . . 3 |
11 | ffun 5334 | . . . . 5 | |
12 | resres 4890 | . . . . . 6 | |
13 | funrel 5199 | . . . . . . 7 | |
14 | resdm 4917 | . . . . . . 7 | |
15 | reseq1 4872 | . . . . . . 7 | |
16 | 13, 14, 15 | 3syl 17 | . . . . . 6 |
17 | 12, 16 | eqtr3id 2211 | . . . . 5 |
18 | 7, 11, 17 | 3syl 17 | . . . 4 |
19 | 18 | feq1d 5318 | . . 3 |
20 | 10, 19 | mpbid 146 | . 2 |
21 | inss2 3338 | . . 3 | |
22 | elpm2r 6623 | . . 3 | |
23 | 21, 22 | mpanr2 435 | . 2 |
24 | 3, 4, 20, 23 | syl21anc 1226 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 crab 2446 cvv 2721 cin 3110 wss 3111 cpw 3553 cxp 4596 cdm 4598 cres 4600 wrel 4603 wfun 5176 wf 5178 (class class class)co 5836 cpm 6606 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pm 6608 |
This theorem is referenced by: lmres 12795 |
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