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Mirrors > Home > ILE Home > Th. List > mapsncnv | Unicode version |
Description: Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | |
mapsncnv.b | |
mapsncnv.x | |
mapsncnv.f |
Ref | Expression |
---|---|
mapsncnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 6564 | . . . . . . . . 9 | |
2 | mapsncnv.x | . . . . . . . . . 10 | |
3 | 2 | snid 3556 | . . . . . . . . 9 |
4 | ffvelrn 5553 | . . . . . . . . 9 | |
5 | 1, 3, 4 | sylancl 409 | . . . . . . . 8 |
6 | eqid 2139 | . . . . . . . . 9 | |
7 | mapsncnv.b | . . . . . . . . 9 | |
8 | 6, 7, 2 | mapsnconst 6588 | . . . . . . . 8 |
9 | 5, 8 | jca 304 | . . . . . . 7 |
10 | eleq1 2202 | . . . . . . . 8 | |
11 | sneq 3538 | . . . . . . . . . 10 | |
12 | 11 | xpeq2d 4563 | . . . . . . . . 9 |
13 | 12 | eqeq2d 2151 | . . . . . . . 8 |
14 | 10, 13 | anbi12d 464 | . . . . . . 7 |
15 | 9, 14 | syl5ibrcom 156 | . . . . . 6 |
16 | 15 | imp 123 | . . . . 5 |
17 | fconst6g 5321 | . . . . . . . . 9 | |
18 | 2 | snex 4109 | . . . . . . . . . 10 |
19 | 7, 18 | elmap 6571 | . . . . . . . . 9 |
20 | 17, 19 | sylibr 133 | . . . . . . . 8 |
21 | vex 2689 | . . . . . . . . . . 11 | |
22 | 21 | fvconst2 5636 | . . . . . . . . . 10 |
23 | 3, 22 | mp1i 10 | . . . . . . . . 9 |
24 | 23 | eqcomd 2145 | . . . . . . . 8 |
25 | 20, 24 | jca 304 | . . . . . . 7 |
26 | eleq1 2202 | . . . . . . . 8 | |
27 | fveq1 5420 | . . . . . . . . 9 | |
28 | 27 | eqeq2d 2151 | . . . . . . . 8 |
29 | 26, 28 | anbi12d 464 | . . . . . . 7 |
30 | 25, 29 | syl5ibrcom 156 | . . . . . 6 |
31 | 30 | imp 123 | . . . . 5 |
32 | 16, 31 | impbii 125 | . . . 4 |
33 | mapsncnv.s | . . . . . . 7 | |
34 | 33 | oveq2i 5785 | . . . . . 6 |
35 | 34 | eleq2i 2206 | . . . . 5 |
36 | 35 | anbi1i 453 | . . . 4 |
37 | 33 | xpeq1i 4559 | . . . . . 6 |
38 | 37 | eqeq2i 2150 | . . . . 5 |
39 | 38 | anbi2i 452 | . . . 4 |
40 | 32, 36, 39 | 3bitr4i 211 | . . 3 |
41 | 40 | opabbii 3995 | . 2 |
42 | mapsncnv.f | . . . . 5 | |
43 | df-mpt 3991 | . . . . 5 | |
44 | 42, 43 | eqtri 2160 | . . . 4 |
45 | 44 | cnveqi 4714 | . . 3 |
46 | cnvopab 4940 | . . 3 | |
47 | 45, 46 | eqtri 2160 | . 2 |
48 | df-mpt 3991 | . 2 | |
49 | 41, 47, 48 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wcel 1480 cvv 2686 csn 3527 copab 3988 cmpt 3989 cxp 4537 ccnv 4538 wf 5119 cfv 5123 (class class class)co 5774 cmap 6542 |
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-reu 2423 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-mpt 3991 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-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 |
This theorem is referenced by: mapsnf1o2 6590 mapsnf1o3 6591 |
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