Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6660 | . . . . . . . . 9 | |
2 | mapsncnv.x | . . . . . . . . . 10 | |
3 | 2 | snid 3620 | . . . . . . . . 9 |
4 | ffvelcdm 5641 | . . . . . . . . 9 | |
5 | 1, 3, 4 | sylancl 413 | . . . . . . . 8 |
6 | eqid 2175 | . . . . . . . . 9 | |
7 | mapsncnv.b | . . . . . . . . 9 | |
8 | 6, 7, 2 | mapsnconst 6684 | . . . . . . . 8 |
9 | 5, 8 | jca 306 | . . . . . . 7 |
10 | eleq1 2238 | . . . . . . . 8 | |
11 | sneq 3600 | . . . . . . . . . 10 | |
12 | 11 | xpeq2d 4644 | . . . . . . . . 9 |
13 | 12 | eqeq2d 2187 | . . . . . . . 8 |
14 | 10, 13 | anbi12d 473 | . . . . . . 7 |
15 | 9, 14 | syl5ibrcom 157 | . . . . . 6 |
16 | 15 | imp 124 | . . . . 5 |
17 | fconst6g 5406 | . . . . . . . . 9 | |
18 | 2 | snex 4180 | . . . . . . . . . 10 |
19 | 7, 18 | elmap 6667 | . . . . . . . . 9 |
20 | 17, 19 | sylibr 134 | . . . . . . . 8 |
21 | vex 2738 | . . . . . . . . . . 11 | |
22 | 21 | fvconst2 5724 | . . . . . . . . . 10 |
23 | 3, 22 | mp1i 10 | . . . . . . . . 9 |
24 | 23 | eqcomd 2181 | . . . . . . . 8 |
25 | 20, 24 | jca 306 | . . . . . . 7 |
26 | eleq1 2238 | . . . . . . . 8 | |
27 | fveq1 5506 | . . . . . . . . 9 | |
28 | 27 | eqeq2d 2187 | . . . . . . . 8 |
29 | 26, 28 | anbi12d 473 | . . . . . . 7 |
30 | 25, 29 | syl5ibrcom 157 | . . . . . 6 |
31 | 30 | imp 124 | . . . . 5 |
32 | 16, 31 | impbii 126 | . . . 4 |
33 | mapsncnv.s | . . . . . . 7 | |
34 | 33 | oveq2i 5876 | . . . . . 6 |
35 | 34 | eleq2i 2242 | . . . . 5 |
36 | 35 | anbi1i 458 | . . . 4 |
37 | 33 | xpeq1i 4640 | . . . . . 6 |
38 | 37 | eqeq2i 2186 | . . . . 5 |
39 | 38 | anbi2i 457 | . . . 4 |
40 | 32, 36, 39 | 3bitr4i 212 | . . 3 |
41 | 40 | opabbii 4065 | . 2 |
42 | mapsncnv.f | . . . . 5 | |
43 | df-mpt 4061 | . . . . 5 | |
44 | 42, 43 | eqtri 2196 | . . . 4 |
45 | 44 | cnveqi 4795 | . . 3 |
46 | cnvopab 5022 | . . 3 | |
47 | 45, 46 | eqtri 2196 | . 2 |
48 | df-mpt 4061 | . 2 | |
49 | 41, 47, 48 | 3eqtr4i 2206 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wceq 1353 wcel 2146 cvv 2735 csn 3589 copab 4058 cmpt 4059 cxp 4618 ccnv 4619 wf 5204 cfv 5208 (class class class)co 5865 cmap 6638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-map 6640 |
This theorem is referenced by: mapsnf1o2 6686 mapsnf1o3 6687 |
Copyright terms: Public domain | W3C validator |