Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6545 | . . . 4 | |
2 | 1 | elmpocl1 5969 | . . 3 |
3 | 2 | adantl 275 | . 2 |
4 | simpl 108 | . 2 | |
5 | elpmi 6561 | . . . . . 6 | |
6 | 5 | simpld 111 | . . . . 5 |
7 | 6 | adantl 275 | . . . 4 |
8 | inss1 3296 | . . . 4 | |
9 | fssres 5298 | . . . 4 | |
10 | 7, 8, 9 | sylancl 409 | . . 3 |
11 | ffun 5275 | . . . . 5 | |
12 | resres 4831 | . . . . . 6 | |
13 | funrel 5140 | . . . . . . 7 | |
14 | resdm 4858 | . . . . . . 7 | |
15 | reseq1 4813 | . . . . . . 7 | |
16 | 13, 14, 15 | 3syl 17 | . . . . . 6 |
17 | 12, 16 | syl5eqr 2186 | . . . . 5 |
18 | 7, 11, 17 | 3syl 17 | . . . 4 |
19 | 18 | feq1d 5259 | . . 3 |
20 | 10, 19 | mpbid 146 | . 2 |
21 | inss2 3297 | . . 3 | |
22 | elpm2r 6560 | . . 3 | |
23 | 21, 22 | mpanr2 434 | . 2 |
24 | 3, 4, 20, 23 | syl21anc 1215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 crab 2420 cvv 2686 cin 3070 wss 3071 cpw 3510 cxp 4537 cdm 4539 cres 4541 wrel 4544 wfun 5117 wf 5119 (class class class)co 5774 cpm 6543 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pm 6545 |
This theorem is referenced by: lmres 12417 |
Copyright terms: Public domain | W3C validator |