![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mapsnf1o2 | Unicode version |
Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s |
![]() ![]() ![]() ![]() ![]() ![]() |
mapsncnv.b |
![]() ![]() ![]() ![]() |
mapsncnv.x |
![]() ![]() ![]() ![]() |
mapsncnv.f |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
mapsnf1o2 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2755 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | mapsncnv.x |
. . . 4
![]() ![]() ![]() ![]() | |
3 | 1, 2 | fvex 5550 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | mapsncnv.f |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | fnmpti 5359 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | mapsncnv.s |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 2 | snex 4200 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
8 | 6, 7 | eqeltri 2262 |
. . . 4
![]() ![]() ![]() ![]() |
9 | vex 2755 |
. . . . 5
![]() ![]() ![]() ![]() | |
10 | 9 | snex 4200 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
11 | 8, 10 | xpex 4756 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | mapsncnv.b |
. . . 4
![]() ![]() ![]() ![]() | |
13 | 6, 12, 2, 4 | mapsncnv 6713 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 11, 13 | fnmpti 5359 |
. 2
![]() ![]() ![]() ![]() ![]() |
15 | dff1o4 5484 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 5, 14, 15 | mpbir2an 944 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-map 6668 |
This theorem is referenced by: mapsnf1o3 6715 |
Copyright terms: Public domain | W3C validator |