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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 5324 | . . 3 |
3 | 1 | rnmpt 4859 | . . . 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 5202 | . . 3 | |
13 | 2, 11, 12 | sylanbrc 415 | . 2 |
14 | fimacnv 5625 | . . . 4 | |
15 | 1 | mptpreima 5104 | . . . 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 4050 ccnv 4610 crn 4612 cima 4614 wfn 5193 wf 5194 |
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 4107 ax-pow 4160 ax-pr 4194 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 |
This theorem is referenced by: f1ompt 5647 fmpti 5648 fvmptelrn 5649 fmptd 5650 fmptdf 5653 rnmptss 5657 f1oresrab 5661 idref 5736 f1mpt 5750 f1stres 6138 f2ndres 6139 fmpox 6179 fmpoco 6195 iunon 6263 mptelixpg 6712 dom2lem 6750 uzf 9490 pcmptcl 12294 upxp 13066 txdis1cn 13072 cnmpt11 13077 cnmpt21 13085 fsumcncntop 13350 cncfmpt1f 13378 mulcncflem 13384 mulcncf 13385 cnmptlimc 13437 sincn 13484 coscn 13485 |
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