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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 5322 | . . 3 |
3 | 1 | rnmpt 4857 | . . . 4 |
4 | r19.29 2607 | . . . . . . 7 | |
5 | eleq1 2233 | . . . . . . . . 9 | |
6 | 5 | biimparc 297 | . . . . . . . 8 |
7 | 6 | rexlimivw 2583 | . . . . . . 7 |
8 | 4, 7 | syl 14 | . . . . . 6 |
9 | 8 | ex 114 | . . . . 5 |
10 | 9 | abssdv 3221 | . . . 4 |
11 | 3, 10 | eqsstrid 3193 | . . 3 |
12 | df-f 5200 | . . 3 | |
13 | 2, 11, 12 | sylanbrc 415 | . 2 |
14 | fimacnv 5623 | . . . 4 | |
15 | 1 | mptpreima 5102 | . . . 4 |
16 | 14, 15 | eqtr3di 2218 | . . 3 |
17 | rabid2 2646 | . . 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 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 crab 2452 wss 3121 cmpt 4048 ccnv 4608 crn 4610 cima 4612 wfn 5191 wf 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 |
This theorem is referenced by: f1ompt 5645 fmpti 5646 fvmptelrn 5647 fmptd 5648 fmptdf 5651 rnmptss 5655 f1oresrab 5659 idref 5734 f1mpt 5748 f1stres 6136 f2ndres 6137 fmpox 6177 fmpoco 6193 iunon 6261 mptelixpg 6710 dom2lem 6748 uzf 9483 pcmptcl 12287 upxp 13031 txdis1cn 13037 cnmpt11 13042 cnmpt21 13050 fsumcncntop 13315 cncfmpt1f 13343 mulcncflem 13349 mulcncf 13350 cnmptlimc 13402 sincn 13449 coscn 13450 |
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