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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 5219 | . . 3 |
3 | 1 | rnmpt 4757 | . . . 4 |
4 | r19.29 2546 | . . . . . . 7 | |
5 | eleq1 2180 | . . . . . . . . 9 | |
6 | 5 | biimparc 297 | . . . . . . . 8 |
7 | 6 | rexlimivw 2522 | . . . . . . 7 |
8 | 4, 7 | syl 14 | . . . . . 6 |
9 | 8 | ex 114 | . . . . 5 |
10 | 9 | abssdv 3141 | . . . 4 |
11 | 3, 10 | eqsstrid 3113 | . . 3 |
12 | df-f 5097 | . . 3 | |
13 | 2, 11, 12 | sylanbrc 413 | . 2 |
14 | 1 | mptpreima 5002 | . . . 4 |
15 | fimacnv 5517 | . . . 4 | |
16 | 14, 15 | syl5reqr 2165 | . . 3 |
17 | rabid2 2584 | . . 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 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 crab 2397 wss 3041 cmpt 3959 ccnv 4508 crn 4510 cima 4512 wfn 5088 wf 5089 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: f1ompt 5539 fmpti 5540 fvmptelrn 5541 fmptd 5542 fmptdf 5545 rnmptss 5549 f1oresrab 5553 idref 5626 f1mpt 5640 f1stres 6025 f2ndres 6026 fmpox 6066 fmpoco 6081 iunon 6149 mptelixpg 6596 dom2lem 6634 uzf 9285 upxp 12352 txdis1cn 12358 cnmpt11 12363 cnmpt21 12371 fsumcncntop 12636 cncfmpt1f 12664 mulcncflem 12670 mulcncf 12671 cnmptlimc 12723 sincn 12769 coscn 12770 |
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