Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fmpt | Unicode version |
Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmpt.1 |
Ref | Expression |
---|---|
fmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpt.1 | . . . 4 | |
2 | 1 | fnmpt 5244 | . . 3 |
3 | 1 | rnmpt 4782 | . . . 4 |
4 | r19.29 2567 | . . . . . . 7 | |
5 | eleq1 2200 | . . . . . . . . 9 | |
6 | 5 | biimparc 297 | . . . . . . . 8 |
7 | 6 | rexlimivw 2543 | . . . . . . 7 |
8 | 4, 7 | syl 14 | . . . . . 6 |
9 | 8 | ex 114 | . . . . 5 |
10 | 9 | abssdv 3166 | . . . 4 |
11 | 3, 10 | eqsstrid 3138 | . . 3 |
12 | df-f 5122 | . . 3 | |
13 | 2, 11, 12 | sylanbrc 413 | . 2 |
14 | 1 | mptpreima 5027 | . . . 4 |
15 | fimacnv 5542 | . . . 4 | |
16 | 14, 15 | syl5reqr 2185 | . . 3 |
17 | rabid2 2605 | . . 3 | |
18 | 16, 17 | sylib 121 | . 2 |
19 | 13, 18 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 crab 2418 wss 3066 cmpt 3984 ccnv 4533 crn 4535 cima 4537 wfn 5113 wf 5114 |
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-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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 |
This theorem is referenced by: f1ompt 5564 fmpti 5565 fvmptelrn 5566 fmptd 5567 fmptdf 5570 rnmptss 5574 f1oresrab 5578 idref 5651 f1mpt 5665 f1stres 6050 f2ndres 6051 fmpox 6091 fmpoco 6106 iunon 6174 mptelixpg 6621 dom2lem 6659 uzf 9322 upxp 12430 txdis1cn 12436 cnmpt11 12441 cnmpt21 12449 fsumcncntop 12714 cncfmpt1f 12742 mulcncflem 12748 mulcncf 12749 cnmptlimc 12801 sincn 12847 coscn 12848 |
Copyright terms: Public domain | W3C validator |